U.M.R. Permutations and Combinations 1 U M R Page 1 Section:I Section I contains multiple choice questions. Each question has 4 choices A), B), C) and D), out of which only one is correct. 01. In a seven digit number only 2 and 3 will present. If no 2’s are consecutive, then the number of such numbers is (A) 26 (B) 33 (C) 32 (D) 53 02. In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70, then the number of diagonals of the polygon is (A) 8 (B) 20 (C) 28 (D) 24 03. The number of positive integers not greater then 100 which are not divisible by 2, 3 or 5 is (A) 26 (B) 36 (C) 30 (D) 42 04. If nk ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ represents the combination of n different things taken k at a time then the value of 100 99 98 3 2 ..... 98 97 96 1 0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + + + + + = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ A) 15050 B) 101000 C) 151500 D) 166650 05. Sum of the proper divisors of 9900 is (A) 33852 B) 23952 C) 23951 D) 33851 06. The number of ways in which 5 ladies and 7 gentlemen can be seated in a round table so that no two ladies sit together is A) 2 7 (720) 2 B) 2 7(360) C) 2 7(720) D) 720 07. A factor P of 10000000099. lies between 9000 and 10,000. The sum of its digits is a) 11 b) 13 c) 17 d) 19 08. Let a i j k = + + r be a vector such that . , . , . r i r j r k r r ur r r r are positive integers. If . r a r r 12 ≤ , then the number of vectors rr is A) 12 9 1 c − B) 12 9 c C ) 12 4 1 c − D) 12 5 1 c − 09. In a polygon of n sides let, N be the number of diagonals. If 10 > − n N , then the least value of n is (A) 7 (B) 8 (C) 9 (D) 10 10. 15 pigeons are suppose to rest in 5 holes . In how many ways they can rest in holes such that each hole should contain at least one pigeon. (A) 16 5 c (B) 14 4 c (C)10 5 c (D) 9 6 c 11. The number of zero’s at the end of 60! Is A) 14 B) 15 C) 16 D) 10 12. There are 4 white 8 blue and 9 yellow shirts. The least number of shirts that can be picked at random so that there are 7 shirts of the same colour is A) 16 B) 17 C) 12 D) 36 13. 20 candidates are sitting at a round table. One has to select 5 of them so that no two of them sitting side by side are selected. Then the number of ways of selecting the candidates is a) 6 15C b) 5 13C c) 6 14C d) ( ) 14 4 4 c 14. There are 12 girls and 15 boys. Out of them we want to have 4 pairs (each pair contain one boy and one girl) for a dance programme. Number of ways this can be done is a).12 15 4 4 C C b).12 15 4 4 P C c). 12 15 4 48 C C d). 15 12 4 44 4 C C 15. Let 1 2 3 x ,x ,x be integers greater than 1. If = 4 5 3 1 2 3 x .x .x 2 .3 .5 , then the possible number of the ordered triple( ) 1 2 3 x ,x ,x is a) 3150 b) 2790 c) 2793 d) 958 16. The number of ways in which 30 coins of one rupee each be given to six persons, so that none of them receives less than 4 rupees is a) 231 b) 462 c) 693 d) 924 17. Two candidate A and B contested for presidentship in a panchayat elections. Each of them got 5 votes. The number of ways a counting be done so that at no stage of counting A lags behind B are A) 5 1 10 6 C B) 5 1 8 5 C C) 5 1 10 5 C D) None 18. The possible number of ordered triples (m,n,p) such that 1 100, 1 50, 1 25 2 2 2 ≤ ≤ ≤ ≤ ≤ ≤ + + m n p m n p and is divisible by 3 is (A) 31250 (B) 30000 (C) 31249 (D) 32150 19. There are 4 identical red strips , 3 identical blue strips and 2 identical white strips . The number of flags with three strips in order can be formed using these strips are U.M.R. Permutations and Combinations 2 U M R Page 2 (A) 10 (B) 15 (C) 20 (D) 25 20. Find the number of different ways in which 13 distinct objects can be divided into two groups of 5 and 8 (a) 1287 (b) 1286 (c) 1280 (d) 1387 21. The number ways of a mixed double game can be arranged from amongst 9 couples if no husband and wife play in the same game is (a) 756 (b) 1512 (c) 3024 (d) 3000 22. The letters of the word MIRROR are arranged in all possible ways these words are written as in a dictionary then the rank of word MIRROR will be A) 23 B) 24 C) 25 D) 26 23. The number of six digit numbers in which digits are ascending order (a) 48 (b) 84 (c) 120 (d) 126 24. The value of Expression 0 1 2 ( 1) ( 2) ......( ) r c C C C x n n n n r = + + + + + + is (a)( 1)r C n r + + (b) 1 r C n + (c) n r C n − (d) None 25. Two numbers ‘a’ & ‘b’ are chosen from the set of {1,2,3……3n}. In how many ways can these integers be selected such that 2 2 a b − is divisible by 3 a) ( ) 2 3 1 2n n n + + b) ( ) 2 3 1 2n n n − + c) ( ) 2 1 1 2n n n + − d) ( ) 2 1 1 2n n n − + 26. The number of distinct rational numbers of the form p/q, where { } , 1, 2,3, 4,5,6 p q∈ is a) 23 b) 32 c) 36 d) 28 27. Let d1, d2, ……, dk be all the divisors of a positive integer n including 1 and n. Suppose d1 + d2 + … + dk = 72. Then the value of 1 2 1 1 1 ...... k d d d + + + a) is 2 72 k b) is 72 k c) is 72 n d) cannot be computed from the given information 28. There are 10 stations on a circular path. A train has to stop at 3 stations such that no two stations are adjacent. The number of such selections must be a) 50 b) 84 c) 126 d) None of these 29. Let n and k be positive integers such that ( ) 1 2 k k n + ≥ . The number of solution (x1, x2, … xk), 1 2 1, 2,..., k x x x k ≥ ≥ ≥ , all integers, satisfying x1 + x2 + … + xk = n, is 2 2 2. 2 n k k m⎛ ⎞ − + − = ⎜ ⎟ ⎝ ⎠ a) mCk b) m-1Ck c) mCk-1 d) Zero 30. An n-digit number is a positive integer with exactly n-digits. Nine hundred distinct n-digit numbers are to be formed by using the digit 2, 5 and 7 only. The smallest value of n for which this is possible is a) 6 b) 7 c) 8 d) 9 31. How many ways are there to form a three-letter sequence using the letters , , , , , a b c d e f containing e when repetition of the letters is allowed a) 90 b) 91 c) 92 d) 89 32. How many integers between l and 10,000 has exactly one 8 and one 9 a) 4 3 × b) 4 3 8 7 × × × c) 2 2 4 3 8 × × × d) 2 4 3 8 × × 33. How many times is the digit 5 written when listing all numbers from 1 to 1,00,000 ? a) 4 5 10 × b) 1 10 100 1000 10,000 + + + + c) 3 5 10 × d) 1 10 100 1000 + + + 34. Number of arrangements of SYSTEMATIC in which each S is immediately followed by a vowel a) 8 3 2 6 C P b) 8 2 4 3 P × × c) 4 8 2 6 P P × d) 8 3 2 6 P P 35. Let N be the number of 7-digit numbers the sum of whose digits is even. The number of +ve divisors of N is a) 64 b) 72 c) 88 d) 126 36. There are 15 different apples and 10 different pears. How many ways are there for Jack to pick an apple or a pear and then Jill to pick an apple and a pear. a) 23 150 × b) 33 150 × c) 43 150 × d) 53 150 × 37. Among the 8! permutations of the digits 1,2,3…..8, consider those arrangements which have the following property. If you take any five consecutive positions, the product of the digits in those positions is divisible by 5. The number of such arrangements is a) 7! b) 2.7! c) 8.7! d) 5!3!4! 38. Let { } 0,1, 2,3,...9 A = be a set consisting of different digits. The number of ways in which a nine digit U.M.R. Permutations and Combinations 3 U M R Page 3 number can be made in which,1 and 2 are present and 1 is always ahead of 2 and repetition of digits is not allowed. a) 65 7! 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ b) 65 9! 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ c) 65 8! 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ d) 65 10! 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 39. Number of points having position vector ai b j c k −+ + when a, b, c ∈ { } 1, 2,3, 4,5 such that 2 3 5 a b c + + is divisible by 4 is a) 140 b) 70 c) 100 d) 120 40. A six digit number is formed using all the six digits 2,3,4,5,7,8, then number of such digits that are divisible by 11 is … a) 36 b) 720 c) 180 d) 72 41. Let N be a natural number if its first digit (from the left) is deleted, it gets reduced to 57 N . The sum of all the digits of N is … a) 15 b) 18 c) 24 d) 30 42. An unlimited number of coupons bearing the letters A, B and C are available, then the number of ways of choosing 10 of these coupons so that they can’t used to spell BAC a) ( ) 10 3 2 1 − b) ( ) 10 2 3 1 − c) 10 2 1 − d) 10 2 43. The integers from 1 to 1000 are written in order around a circle. Starting at 1, every fifteenth numbers is marked (ie. 1,16,31 etc). This process is continued until a number is reached which has already been marked, then unmarked numbers are …. A) 200 B) 400 C) 600 D) 800 44. Five distinct letters are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the two letters with at least three between every two. The total number of ways in which this can be done is . A) 1200 B) 1800 C) 2400 D) 3000 45. If 2009 10 10 K K C = ∑ simplifies to n P C , where p is prime, then ( ) n p + has the value equal to A) 2018 B) 2019 C) 2020 D) 2021 46. In a seven digit number only 2 and 3 will present. If no 2’s are consecutive, then the number of such numbers is (A) 26 (B) 33 (C) 32 (D) 53 47. In a polygon of n sides let, N be the number of diagonals. If 10 > − n N , then the least value of n is (A) 7 (B) 8 (C) 9 (D) 10 48. The number of ways in which 2 rupee coins, 3 two rupee coins and 4 five rupee coins can be arranged in a row so that at least one coin is separated from the coins of same value a) 1260 b) 1254 c) 1257 d) 1258 49. Number of ways in which the number 44100 can be resolved as a product of two factors which are relatively prime is a) 7 b) 15 c) 8 d) 16 50. Sum of all the odd divisors of 360 is A) 70 B) 78 C) 80 D) 88 51. The number of points (x, y, z) in space whose each co-ordinate is a negative integer such that x+y+z+12 = 0 is A) 55 B) 60 C) 65 D) 70 52. The no. of natural numbers between 100 and 1000 in which digits are in strictly increasing order from left to right. (A) 120 (B) 720 (C) 84 (D) 504 53. The number of three digit numbers, whose middle digit is a prime number and unit digit is from the set {0, 3, 6, 9}, is (A) 81 (B) 100 (C) 121 (D) 144 54. The maximum power of 7, present in 2.4.6.8 . . . 998. 1000, is A) 82 (B)92 (C)102 (D) 81 55. The number of integral solutions of the equation 1 2 3 4 5 . . . . 2310 x x x x x = are A) 55 B) 5 6.5 C) 5 16.5 D) 6 5 56. Let N = a a a a a a be a 6 digit number (all digits repeated) and let , a β be the roots of the equation 2 11 0 x x λ − + = , then product of all possible values of λ is A) 1628 B) 672 C) 624 D) 632 57. There are 12 pairs of shoes in a box. Then the possible number of ways of picking 7 shoes so that there are exactly two pairs of shoes are A) 63360 B) 63300 C) 63260 D) 63060 58. The number of different ordered triplets (a, b, c), a, b, c∈I such that these can represent sides of a U.M.R. Permutations and Combinations 4 U M R Page 4 triangle whose perimeter is 21, is A) 12 B) 31 C) 55 D) 91 59. The number of different permutations of all the letters of the word 'PERMUTATION' such that any two consecutive letters in the arrangement are neither both vowels nor both identical is A) 63 × 6! × 5! B) 57 × 5! × 5! C) 33 × 6! × 5! D) 7 × 7! × 5! 60. The number of different words that can be formed using all the letters of the word 'SHASHANK' such that in any word the vowels are separated by atleast two consonants, is A) 2700 B) 1800 C) 900 D) 600 61. A delegation of four students is to be selected from a total of 12 students. No.of ways in which the delegation be selected if two particular students refuse to be together and two other particular students wish to be together only A) 255 B) 226 C) 210 D) 202 62. No. of even divisors of 10800 A) 48 B) 47 C) 57 D) 59 63. In a certain test there are n questions. In this test 2n i − students gave wrong answers to at least i question, where i =1,2,-----,n. If the total no.of Wrong answers given is 2047, then n is equal to A) 10 B) 11 C) 12 D) 13 64. No.of rectangles excluding squares from a rectangle of size 7x 4 A) 220 B) 216 C) 208 D) 202 65. There are 4 white 8 blue and 9 yellow shirts. The least number of shirts that can be picked at random so that there are 7 shirts of the same colour is A) 16 B) 17 C) 12 D) NONE 66. 20 candidates are sitting at a round table. One has to select 5 of them so that no two of them sitting side by side are selected. Then the number of ways of selecting the candidates is A) 6 15C B) 5 13C C) 6 14C D) None of these 67. The exponent of 12 in 100! Is A) 48 B) 49 C) 96 D) none of these 68. How many different 9 digit numbers can be formed from the number 22 33 55 888 by rearranging its digits so that the odd digits occupy even positions A) 16 B) 36 C) 60 D) 180 69. Total number of even divisors of 189000 that are divisible by 15, are A) 128 B) 54 C) 27 D) 72 70. If r,s,t are distinct prime numbers and p,q are the positive integers such that LCM of p,q is r2t4s2, then the number of ordered pairs (p,q) is A) 224 B) 225 C) 254 D) 252 71. The number of three digit numbers of the form xyz such that x < y and z ≤ y is A) 176 B) 278 C) 276 D) 240 72. Number of ways of factorising 2 3 1 4 5 1 2 .3 .5 .7 .11 .13 into two factors m and n such that m and n are relatively prime is A) 31 B) 62 C) 30 D) None of these 73. The number of divisors of 9600 including 1 and 9600 is A) 60 B) 58 C) 48 D) 46 74. A round table conference is to be held between 20 delegates of 2 countries. In how many ways can they be seated if two particular delegates are always to sit together A) 2 (18!) B) 19! -2(18!) C) 19! D) 18! 75. The number of divisors of 441, 1125 and 384 are in A) A.P. B) G.P. C) H.P. D) none of these 76. Number of ways to select two distinct natural numbers from {1,2,....,100} such that their product is divisible by 3 is A)3300 B)3267 C)1067 D)None of these 77. Number of 5 digit numbers of distinct digits and whose middle term is largest is A) 8 3 3 6 . n n n C = ∑ B) 8 3 3 n n C = ∑ C) 8 3 3 2 . n n n C = ∑ D) None of these 78. Three like red roses and three like pink roses are used to form a garland. The number of ways is A) 3 B) 4 C) 6 D) None of these 79. 3 men and 6 women are to be seated along a round table. Number of cases where? A) 1140 B) 1440 C) 780 D) None of these 80. Let A be the set of 4-digit numbers a1a2a3a4 where a1 > a2 > a3 >a4 , then n(A) is equal to A) 126 B) 84 C) 210 D) none of these 81. There are four pairs of shoes of different sizes. Each of the 8 shoes can be coloured with one of the four colours : Black, Brown, White and Red. The number of ways the shoes can be coloured so that in atleast three pairs, the left shoe and the right shoe do not have the same colour is A) 4 12 B) 3 28 12 × C) 3 16 12 × D) 3 4 12 × U.M.R. Permutations and Combinations 5 U M R Page 5 82. The number of ways in which 6 boys and 6 girls can be seated at a round table so that no two girls sit together and two particulars girls do not sit next to a particular boy is A) 6. 4 B) 2. 5. 4 C) 2. 6. 4 D) 5. 4 83. How many different anagrams can be made by using the letters of the word ‘ALABAMA’ A) 210 B) 5040 C) 181440 D) 15120 84. A password consists 3 letters followed by two digits. How many passwords can be Made if no letter and digit is repeated. A) 140400 B) 1404000 C) 14040 D) 14400 85. Number of zeros at the end of (127)! A) 31 B) 30 C) 10 D) 0 86. The number of solutions of 51 x y z + + = such that x,y,z are odd natural numbers. A) 325 B) 300 C) 330 D) 350 87. The number of ways in which a mixed double game can be arranged for 7 couples such that no husband and his wife are allowed to play in the same game. A) 420 B) 1512 C) 840 D) 120 88. When simplified, the expression 5 47 52 4 3 1 n n C C− = + ∑ equals A) 47 5 C B) 49 4 C C) 52 5 C D) 52 4 C 89. The maximum number of points into which 4 circles and 4 straight lines intersect, is: A) 26 B) 50 C) 56 D) 72 90. The sides , AB BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is: A) 205 B) 208 C) 220 D) 380 91. The number of selections of four letters from the letters of the word ASSASSINATION is A) 72 B) 71 C) 66 D) 52 92. Given that n is odd, the number of ways in which three numbers in AP can be selected from 1, 2, 3, 4, ……., n is A) ( )2 1 2 n − B) ( )2 1 4 n + C) ( )2 1 2 n + D) ( )2 1 4 n − 93. A is a set containing n elements. A subset 1 P is chosen and A is reconstructed by replacing the elements of 1 P . The same process is repeated for subsets 2 3 , ,.......... m P P P with 1 m > . The number of ways of choosing 1 2 , ,......... m P P P , so that 1 2 ....... m P P P A ∪ ∪ ∪ = is A) ( ) 2 1mn m − B) ( ) 2 1m n − C) m n m C + D) ( ) 2 1n m − 94. The number of ways in which a mixed double game of tennis can be arranged from amongst 9 married couples, if no husband and wife play in the same game is A) 756 B) 1512 C) 3024 D) 2268 95. The number of ways in which we can choose 2 distinct integers from 1 to 100 such that difference between them is at most 10 is A) 10 2 C B) 72 C) 100 90 2 2 C C − D) 100 80 2 2 C C − 96. The number of divisors of 22.33.53.75 of the form 2n + 1, n ∈ N is A) 96 B) 95 C) 94 D) 924 97. The number of ways in which 5 identical balls can be kept in 10 identical boxes, if not more than one can go into a box, is A) 10P5 B) 10 5 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ C) 5 D) 1 98. If n identical dice are rolled simultaneously, the number of distinct throws is A) n+5C5. B) n 6 6 6 n− + C) 6n D) n 6 6 n− 99. If four dice are rolled once the number of ways of getting the sum ‘10’ is A) 76 B) 84 C) 80 D) 60 100. The number of zero’s at the end of 60 ∠ is A) 10 B) 12 C) 14 D) 16 101. The number of 4 digited numbers that can be formed using the digits 0,1,2,3,4,5 which are divisible by 6 when repetition of the digits is allowed is A) 180 B) 190 C) 200 D) 220 102. Let { } 1, 2,3,...100 S = . The number of non-empty subsets A of S such that the product of elements in A is even is A) ( ) 50 50 2 2 1 − B) 100 2 1 − C) 50 2 1 − D) 100 75 2 2 − U.M.R. Permutations and Combinations 6 U M R Page 6 103. Let { } 1, 2,3, 4,..., 20 A = . The number of onto functions from A to A such that ( ) f k is a multiple of 3 whenever k is a multiple of 4 is A) 5. 6. 9 B) 6 5 15 C) 5 6 14 D) 15. 6 104. Let { } 1 2 3 , , 0,1, 2,3,...9 a a a ∈ . Then the number of ordered triads ( ) 1 2 3 , , a a a Satisfying the condition 1 2 3 a a a + + is a multiple of 3 is A) 327 B) 333 C) 334 D) 336 105. There are four pairs of shoes of different sizes. Each of the 8 shoes can be coloured with one of the four colours : Black, Brown, White and Red. The number of ways the shoes can be coloured so that in atleast three pairs, the left shoe and the right shoe do not have the same colour is A) 4 12 B) 3 28 12 × C) 3 16 12 × D) 3 4 12 × 106. The number of ways in which 6 boys and 6 girls can be seated at a round table so that no two girls sit together and two particulars girls do not sit next to a particular boy is A) 6. 4 B) 2. 5. 4 C) 2. 6. 4 D) 5. 4 107. The number of non-negative integral solutions of 1 2 3 16 20 x x x ≤ + + ≤ is A) 112 B) 1050 C) 685 D) 955 108. Let { } 1, 2,3,...100 S = . The number of non-empty subsets A of S such that the product of elements in A is even is A) ( ) 50 50 2 2 1 − B) 100 2 1 − C) 50 2 1 − D) 100 75 2 2 − 109. Let { } 1, 2,3, 4,..., 20 A = . The number of onto functions from A to A such that ( ) f k is a multiple of 3 whenever k is a multiple of 4 is A) 5. 6. 9 B) 6 5 15 C) 5 6 14 D) 15. 6 110. Let { } 1 2 3 , , 0,1, 2,3,...9 a a a ∈ . Then the number of ordered triads ( ) 1 2 3 , , a a a Satisfying the condition 1 2 3 a a a + + is a multiple of 3 is A) 327 B) 333 C) 334 D) 336 111. There are four pairs of shoes of different sizes. Each of the 8 shoes can be coloured with one of the four colours : Black, Brown, White and Red. The number of ways the shoes can be coloured so that in atleast three pairs, the left shoe and the right shoe do not have the same colour is A) 4 12 B) 3 28 12 × C) 3 16 12 × D) 3 4 12 × 112. The number of ways in which 6 boys and 6 girls can be seated at a round table so that no two girls sit together and two particulars girls do not sit next to a particular boy is A) 6. 4 B) 2. 5. 4 C) 2. 6. 4 D) 5. 4 113. The number of non-negative integral solutions of 1 2 3 16 20 x x x ≤ + + ≤ is A) 112 B) 1050 C) 685 D) 955 114. The number of ways of forming an arrangement of 4 letters from the letters of the word “IITJEE” is A) 66 B) 96 C) 102 D) 180 115. The number of four digit numbers that can be formed using the digits 1,2,3,4,5,6 that are divisible by 3,when repetition of digits is allowed, is A) 3 2 2 3 × B) 3 3 2 3 × C) 3 4 2 3 × D) 4 3 2 3 × 116. The letters of the word “DRAWER” are arranged in alphabetical order. The number of arrangements that precede the word “ REWARD” is A) 241 B) 242 C) 247 D) 248 117. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical U.M.R. Permutations and Combinations 7 U M R Page 7 order as in an English dictionary. The number of words that appear before the word COCHIN is A) 360 B) 192 C) 96 D) 48 118. Total number of positive integral solutions of 15 < x1 + x2 + x3 ≤ 20, is equal to A) 1125 B) 1150 C) 1245 D) 685 119. Eight straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. The number of parts in to which these lines divide the plane is A) 29 B) 32 C) 36 D) 37 120. The number of three digit numbers with three distinct digits such that one of the digits is the arithmetic mean of the other two is A) 120 B) 180 C) 112 D) 104 121. The number of ways of forming an arrangement of 5 letters from the letters of the word “IITJEE” is A) 60 B) 96 C) 120 D) 180 122. The number of the functions f from the set X = {1, 2, 3} to the Y = {1, 2, 3, 4, 5, 6, 7} such that f(i) ≤ f(j) for i < j and i, j ∈ X is A) 6C3 B) 7C3 C) 8C3 D) 9C3 123. How many combinations can be made up of 3 hens, 4 ducks and 2 geese so that each combination has hens, ducks and geese? ( birds of same kind all different) A) 305 B) 315 C) 320 D) 325 124. The number of ordered pairs of positive integers (a,b) such that LCM of a&b is 3 7 13 2 5 11 is A) 2385 B) 2835 C) 3825 D) 8325 125. If p P P p p p 3 5 6 1 2 4 15 2 3 5 7 11 13 = then 6 r r 1P = ∑ is A) 24 B) 23 C)22 D)21 126. Let { } { } 1 2 3 7 1 2 3 A x ,x ,x ....x ,B y y y = = . The total number of functions f :A B → that are onto and there are exactly three elements x in A such that ( ) 2 f x y = , is equal to A) 7 2 14. C B) 7 3 14. C C) 7 2 7. C D) 7 3 7. C 127. Number of ways of selecting two integers ‘a’ and ‘b’ from the set { } 1, 2,3 5 , n n N − − − − ∈ so that 4 4 a b − is divisible by 5 A) 2 17 5 2 n n + B) 2 15 17 2 n n − C) 2 17 5 2 n n − D) Section:II 01. For a conference 3 countries have sent 2 delegate each while 3 other countries have sent one delegate each. Number of ways they can be seated in a row so that delegates of the same country are not side by side is x, mark all the correct alter natives for x is a) .244x720 b) 3x8!+760x6!-5! C) 9!-6(8!) + 12 (7!)− 8 (6!) d) None 02. There are 7 pigeons and 7 pigeon holes. They come out when they are disturbed by a gunshot. If m is the number of possible ways they go into the holes one each so that none went to their natural habitat, mark all possible alternatives for m. a) 1854 b) 2 3 4 5 6 7! 7c 6! 7c 5! 7c 4! 7c 3! 7c 2! 7c 1 1 − + − + − + − c) 0 7! 7! 7! 7! 7! 7c 2! 3! 4! 5! 6! − + − + − d) None of these 03. If x is the number of 5 digit numbers sum of whose digits is even and y is the number of 5 digit numbers, sum of whose digit is odd then (a) x = y (b) x + y = 90000 (c) x = 45,000 (d) x < y 04. The number of selection of four letters taken from the word COLLEGE must be (a) 22 (b) 18 (c) Coefficient of 4 x in the expansion of 2 2 3 (1 ) (1 ) x x x + + + (d) 32 05. A class has 30 students. The following prizes are to be awarded to the students of this class first and second in mathematics, first and second in physics, first in chemistry and first in biology, If N denotes the number of ways in which this can be done then, A) 400 | N B) 600 | N C) 8100 | N D) N divisible by 4 distinct prime numbers. 06. Letters of the word SUDESH can be arranged in A) 120 ways when two vowels are together B) 180 ways when two vowels occupy in alphabetical order U.M.R. Permutations and Combinations 8 U M R Page 8 C) 24 ways when vowels and consonants occupy their respective places D) 240 ways when vowels do not occur together 07. The total number of positive integers with distinct digits ( in decimal system) must be A) infinite B)less than 101 10i i= ∑ C) equal to 101 10i i= ∑ D)equal to 9 9 9 9 9 8 9 9 8 7 ...... 9 9 8! + × + × × + × × × + + × × 08. Eight people enter an elevator. At each of four floors atleast one person leaves the elevator after which elevator is empty. The number of ways in which this is possible, is a) ( ) ( ) 4 8 4 0 1 4 i i i C i = − − ∑ b) ( ) ( ) 4 4 8 0 1 8 i i i C i = − − ∑ c) less than 48 d) 8C4 – 1 09. In a certain test i a students gave wrong answers to at least i questions ( ) 1, 2,3.... i k = . No student gave more than k wrong answers, then a) Number of students who gave wrong answer to exactly i questions 1 i i a a− = − b) Number of students who gave wrong answers to exactly i questions 1 i i a a+ = − c) The total no of wrong answers must be 1 2 3 2 3 .... k a a a ka + + + + d) Total no. of wrong answers must be 1 2 .... k a a a + + + . 10. A box contains 4 white balls, 5 black balls and 6 red balls. In how many ways can four balls be drawn from the box if at least one ball of each colour is to be drawn ( If balls of same colour are different) a) 5 6 12 4 1 1 1 1 C C C C b) 1440 c) 720 d) 5 6 12 4 1 1 1 1 1 2 C C C C 11. In how many ways can the letters of the word INTERMEDIATE be arranged so that the order of the vowels as they occur in the given word do not change a) 12 6 6! 2! C b) 12! 3!2! c) ( )2 12 6 6! 3!2! C d) 12! 6!2! 12. The no. of words formed with or without meaning, each of 3 vowels and 2 consonants from the letters of the word INVOLUTE is written in the form of 2 .3 .5 .7 a b c d then a) 6 a = b) 2 b = c) 1 c = d) d=0 13. The sum of all three digited numbers that can be formed from the digits 1 to 9 and when the middle digit is perfect square is a) 1,34,055 (When repetitions are allowed) b) 1,70,555 (When repetitions are allowed) c) 8,73,74 (When repetitions are not allowed) d) 93,387 (When repetitions are not allowed) 14. There are ‘n’ intermediate stations on a railway line from one terminus to another, then number of ways can the train stop at 3 of these intermediate stations: Let x be the number of ways that train stops at ‘3’ consecutive stations, y be the no. of ways that at least two stops are consecutive, z be the no. of ways that no two stations are consecutive A) ( )2 2 y n = − B) 2 x n = − C) 2 3 n z c − = D) 3 2 n z c − = 15. There are 7 pigeons and 7 pigeon holes. They come out when they disturbed by a gunshot, then the number of possible ways they go into the holes one each so that none go to their actual habitate is A) 1854 B) 7 7 7 7 7 7 1 2 3 4 2 6 7 6 5 4 3 2 1 c c c c c c − + − + − + − C) 7 0 7 7 7 7 7 2 3 4 5 6 c − + − + − D) 1 1 1 1 1 1 7 2 3 4 5 6 7 ⎡ ⎤ − + − + − ⎢ ⎥ ⎣ ⎦ 16. The number of ways of selecting two squares ( ) 1 1 × on a chess board having ( ) 8 8 × squares such that A) They have one side in common is 112 B) They have one corner in common is 98 C) They lie on the same diagonal is 280 D) They are of different colours is 1024 17. For the equation 16, x y z w + + + = the no.of positive integral solutions is equal to A) The no.of ways in which 12 identical things can be distributed among 4 persons. B) The no.of ways in which 16 identical things can be distributed among 4 persons. C) coefficient of 16 x in 0 1 2 16 4 ( ) x x x x + + + − − − − − + D) coefficient of 16 x in 1 2 3 16 4 ( ) x x x x + + + − − − − − + 18. If 10! 2 3 5 7 p q r s = , then A) 7 p = B) 4 q = C) 2 r = D) 2 s = 19. Six balls of different colours are to be placed is 3 boxes of different sizes. Each box can hold all the six balls. Number of ways of placing the balls in the boxes so that no box remains empty , is U.M.R. Permutations and Combinations 9 U M R Page 9 A) 6 6 3 3.2 3 − + B) 2 3 3 3 6 6.5 6 4 C − + C) 540 D) ( ) 4 3 6! 6! 6 3! 3!2! 2! 3! C ⎛ ⎞ ⎜ ⎟ + + ⎜ ⎟ ⎝ ⎠ 20. The number of ways of selecting 6 cards from exactly 3 suits out of 4 suits (13 cards in each suit) are A) ( ) ( ) 6 6 6 39 3 26 6 13 C C C − + B) ( ) ( ) 6 6 6 4 39 12 26 12 13 C C C − + C) ( )( ) ( ) 4 3 2 4 2 3 2 12 13 13 24 13 13 13 4 13 C C C C C + + D) ( ) ( ) 4 2 2 2 3 3 13 169 6 13 13 13 13 C C C C + × × + 21. For a conference 3 countries have sent 2 delegates each, while 3 other countries have sent one delegate each. Number of ways they can be seated in a row so that delegates of the same country are not side by side, is A) 244 × 720 B)3 × 8! + 76 × 6! – 5! C) 9! – 6.8 ! + 12 .7 ! – 8.6! D) None 22. There are n faculty members in a university . The faculty assembly consists of r members. Out of r assembly members k of them are selected for senate. The number of ways of selecting assembly members and senate is x. Then all possible values of x are. A) 3 . k C C n n B) r k C C n n + C) r k C C n r D) k rk C C n n k − − 23. Total number of ways in which four boys and four girls can be seated around a round table, so that no two girls sit together, is equal to A) 4!5! B) 3! 4! C) 5(4!)2 D) 4(3!)2. 24. 7 men and 7 women are to sit round a table so that there is a man on either side of a woman. The number of seating arrangement is A) (7!)2/6 B) 7(6!)2 C) 6! 7! D) (7!)2/7 25. If 100! = 2α 3β 5γ 7δ ... , then A) α = 97 B) β = (1/2) (α + 1) C) γ = (1/2) β D) δ = (1/3) β 26. The number of non-negative integral solution of x1 + x2 + x3 + x4 ≤ n (where n is a positive integer) is A) n+3C3 B) n+4C4 C) n+5C5 D) n+4Cn 27. If 4 dice are rolled then the number of ways the score is 6. A) 10 B) 3 5C C) 3 C 6 D) 4 C 6 28. Let { } 1, 2,3,......., S n = . If X denote the set of all subsets of S containing exactly two elements, then the value of ( ) min A X A ∈ ∑ is given by A) 1 3 n C + B) 3 nC C) ( ) 2 1 6 n n − D) ( )( ) 1 2 6 n n n − − 29. A class has 30 students. The following prizes are to be awarded to the students of this class. First and second in Mathematics; first and second in Physics; first in Chemistry and first in Biology. If N denote the number of ways in which this can be done, then A) 400 |N (400 divides N) B) 600 |N (600 divides N) C) 8100 |N (8100 divides N) D) N is divisible by four distinct prime numbers 30. If n objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is A) ( )( )( ) 1 2 3 4 6n n n − − − B) 2 3 n C − C) 3 3 3 2 n n C C − − + D) 1 3 n C − 31. The number of non-negative integral solutions of 1 2 3 4 x x x x n + + + ≤ (where n is a positive integer) is A) 5 n n C + B) 4 4 n C + C) 5 5 n C + D) 4 n n C + 32. Six balls of different colours are to be placed is 3 boxes of different sizes. Each box can hold all the six balls. Number of ways of placing the balls in the boxes so that no box remains empty, is A) 6 6 3 3.2 3 − + B) 2 3 3 3 6 6.5 6 4 C − + C) 540 D) ( ) 4 3 6! 6! 6 3! 3!2! 2! 3! C ⎛ ⎞ ⎜ ⎟ + + ⎜ ⎟ ⎝ ⎠ 33. The number of ways in which we can choose 2 – distinct integers from 1 to 100 such that the difference between them is at most ‘10’ is A) C C 2 2 100 90 − B) C C 98 88 100 90 − C) C C 2 88 100 90 − D) C C 2 88 100 90 + 34. For the equation x + y + z + w = 19, the number of positive integral solution is equal to A) The number of ways in which 15 identical things can be distributed among 4-persons B) The number of ways in which 19-identical things can be distributed among 4-persons C) Coefficient of 19 x in ( )4 0 1 2 19 x x x ..... x + + + + D) Coefficient of 19 x in ( )4 2 3 19 x x x .... x + + + + U.M.R. Permutations and Combinations 10 U M R Page 10 35. If the letters of the word ‘NARAYANA’ are permuted in all possible ways and the words thus formed are arranged in the dictionary order , then the rank of the word ‘NARAYANA’ is not a multiple of A)2 B)3 C)7 D) 5 36. If the number of divisors of the form ‘4K+1’ in 5 7 9 3 5 7 is L then A) L is divisible by 7 B) L is divisible by 8 C) L is divisible by 11 D) L is divisible by 5 37. Thirteen persons are sitting in a row. Number of ways in which four persons can be selected so that no two of them are consecutive is equal to ____ A) number of ways in which all the letters of the word “M A R R I A G E” are permutated if no two vowels are never together. B) number of numbers lying between 100 and 1000 using only the digits 1,2,3,4,5,6,7 without repetition. C) number of ways in which 4 alike chocolates can be distributed among 10 children so that each child getting at most one chocolate. D) number of triangles can be formed by joining 12 points in a plane, of which 5 are collinear. 38. The number of ways in which we can choose 2 distinct integers from 1 to 200 so that the difference between them is atmost 20 is A) 3790 B) 200 180 2 2 C C − C)180 1 19 20 C 20 2× × + D)180 2 C 39. The position vector of a point P is ˆ ˆ ˆ r xi yj zk, = + + r when x, y, z ∈ N and ˆ ˆ ˆ a i j k. = + + r If r.a 10, = r r the number of possible position of P is A) 36 B) 72 C)66 D) 9 2 C Section :III Section III contains Reasoning type questions. Each question contains Statement: 1 and Statement: 2. A) Both the statements are TRUE and Statement: 2 is the correct explanation of Statement: 1 B) Both the statements are TRUE but Statement: 2 is NOT the correct explanation of Statement-1 C)Statement-1 is TRUE and Statement: 2 is FALSE. D) Statement: 1 is FALSE and Statement: 2 is TRUE. 01. Statement-I: 260 when divided by 7 leaves remainder 1 Statement-II:( ) 2 0 1 2 1 ... n n n n n n n x c cx c x cx + = + + + + where n N ∈ Key : A 02. Statement-I: If n is the number of positives integers less then 10,000 which are divisible by all the integers from 2 to 10 (including both), then 5 1 < ≤ n . because Statement-II:The number which is a multiple of two positive integers m and n is also a multiple of the least common multiple of both m and n 03. Statement-I: Highest power of 10 which can divide ( ) 100 ! is 24 Statement-II: According Euler’s concept Highest power of any prime number in ! n can be calculated as follows. Highest power of any prime 2 3 ( ) .... n n n P p p p ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + + + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Key : A 04. Statement-I:20 Identical balls are distributed into 10 boxes in 29 9 c ways Statement-II: Number of positive integral solutions of the equation 1 2 .... n X X X r + + + = are 1 r r n c − − Key : B 05. Statement-I: If n is the number of positives integers less then 10,000 which are divisible by all the integers from 2 to 10 (including both), then 5 1 < ≤ n . because Statement-II:The number which is a multiple of two positive integers m and n is also a multiple of the least common multiple of both m and n Key: A 06. Statement-I: 8 different flags are used to list on 4 flagstaff . If all the flags are used, then it is considered to be a signal. Then the number of ways in which some of the flagstaffs may not have even single flag is 11!. Because Statement-II:Number of ways of distributing n different objects to r persons such that n r 1 n P + − some person may not get even one. Key:D U.M.R. Permutations and Combinations 11 U M R Page 11 07. Statement-I: The number of ways of getting a total of 17 if 4 different dice are cast is 104 . Statement-II: distribution of n identical items to r different boxes is such a way that each box may be given at least one item is n 1 r 1 P − − . Key : C 08. Statement-I:If 8 letters are placed in 8 addressed envelopes, then the number of ways of putting the letters in envelopes so that 3 letters go into correct envelopes and none of the remaining letters go into correct envelop is 2464. Because Statement-II:Number of derangements of n elements from their n habitats is ( )n 1 1 1 1 1 ...... 1 1! 2! 3! n! ⎛ ⎞ − + − + + − ⎜ ⎟ ⎝ ⎠ Key: B 09. Statement-I: Let { /A x = x is a prime number and x<30}. Then the number of different rational numbers. Whose numerator and denominator belong to A is 93. Statement-II: pq is a rational number. 0 q ∀ ≠ and , p q I ∈ Key: D 10. Statement-I: The number of selections of four letters taken from the word PARALLEL must be 15 Because Statement-II: Coefficient of 4 x in the expansion of ( ) 3 1 x − − is 15 ( ) 1 x < Key: D 11. Statement-I: If { } : 1, 2,3, 4,5 {1, 2,3, 4,5} f → then the number of onto functions such that ( ) f i i ≠ is 42 Statement-II: If n things are arranged in row, the number of ways in which they can be de-arranged so that no one of them occupies its original place is ( ) 1 1 1 ! 1 ....... 1 1! 2! ! n n n ⎛ ⎞ − + + − ⎜ ⎟ ⎝ ⎠ Key: D 12. Statement-I: Number of ways of distribution of 12 identical balls into 3 identical boxes is 19 Because Statement-II:Number of ways of distribution of n identical objects among r persons, each one of whom can receive any number of objects is 1 1 r n r c− + − Key: D 13. Statement-I: As n → ∞ , the number of ways of arranging n numbered things in a line where none of them occupies its original position must approach 1/e. Statement-II: 1 1 1 1 1 1 ... 1! 2! 3! 4! e− = − + − + − ∞ Key: D 14. Statement-I: The number of ways of writing 1400 as a product of two positive integers is 12. Statement-II:1400 is divisible by exactly three prime numbers kEY : B 15. Statement –I : Let k d d d d ..... , , 3 2 1 be all the factors of a fixed positive integer n including 1 and n. If 72 ........ 3 2 1 = + + + + k d d d d then the value of n d d d d k 72 1 ........ 1 1 1 3 2 1 = + + + + Because Statement – 2 : For a positive integer n, if d is a factor then dn is also a factor. Key: A 16. Statement -1: No. of ways in which a person in a Buffet dinner could select 3 sweets. When 6 types of sweet are displayed in the dinner is 56. Statement-2: No.of selections of r things from n things when repetition of things is allowed is 1 n r r C + − . Key: A 17. n identical dies are rolled simultaneously. Statement -1: The number of distinct throws is n+5C5. because Statement-2: 6 6 r r 1 C = ∑ n–1Cr–1 = n+5C5 Key: A 18. Statement -1: Total number of different functions from set A having 4 elements to a set B having 2 elements is 16. because U.M.R. Permutations and Combinations 12 U M R Page 12 Statement-2:The number of ways in which m different things can be distributed into n different parcels, blank lots being admissible, is mn. Key: C 19. Statement -1:A selection of 10 balls can be made from an unlimited no.of Red, white, blue and Green balls is 286. Statement-2:No.of terms in the expansion of ( ) 1 2 n x x xr + + − − − − − is 1 1 n r Cr + − − Key: A 20. Statement -1: Let { /A x = x is a prime number and x<30}. Then the number of different rational numbers. Whose numerator and denominator belong to A is 93. Statement-2: pq is a rational number. 0 q ∀ ≠ and , p q I ∈ Key: D 21. Statement-I:The number of ways of partitioning the set { } , , , a b c d into one or more non empty subsets is 16. Because Statement-II: The number of ways of partitioning a set of ( ) m n + members into two subsets of m and n members is m n m+ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ if m ≠ n and 2 12 mm ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ if m = n. Key: D 22. Statement-I: The number of ways of distributing n identical objects in r distinct boxes is 1 1 − − + r r n C . Statement-II: The number of arrangement of n objects of one kind and r – 1 objects of another kind in a line must be ( ) ( )! 1 ! ! 1 −− +r n r n Key: A 23. Statement-I: The number of ways of writing 1400 as a product of two positive integers is 12. Statement-II: 1400 is divisible by exactly three prime numbers. Key: B 24. Statement-I:The expression n! (20 – n)! is minimum when n = 10 Statement-II: r mC 2 is maximum when r = m Key: A 25. Statement – 1 : There are 5 integers between 1 and 200 each of which have 10 divisors . Because Statement – 2 : If p1,p2, p3 are distinct prime numbers then the number of divisors of 2 3 1 1 2 3 p p p α α α is ( )( )( ) 1 2 3 1 1 1 α α α + + + . Key: A 26. Statement – 1 : If 1 100, 1 50, 1 25 m n p ≤ ≤ ≤ ≤ ≤ ≤ ,then possible number of ordered triplets (m, n, p) such that 2 2 2 m n p + + is divisible by 3 is 3125. Because Statement – 2 : 2 2 3, , , a b c a b c N + + ∈ is always divisible by 3 if a, b, c are either odd or even numbers Key: D 27. Statement – 1 : The number of ways of forming a quadrilateral by joining the vertices of n sided regular polygon such that the quadrilatral has exactly odd number of side common with the polygon ( ) 6 n ≥ is 3 2 11 32 2 n n n − + . Because Statement – 2 : There are n straight lines is a plane such that n1 of them are parallel in one direction , n2 are parallel in different direction and so on, nk are parallel another direction such that 1 2 ...... k n n n n + + + = . Also no three of the given lines meet at a point. Then the total number of points of intersection is 2 2 1 12 n r r n n= ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ ∑ . Key: B 28. Statement – 1 : If 12 divides the seven digit number ab 313 ab, then the smallest value of a + b is 4 Because statement – 2 : Any natural number is divisible by 6 if it is divisible by 2 as well as 3. Key: C 29. Statement – 1: The number of ways in which a set of 10boys can be divided into three groups U.M.R. Permutations and Combinations 13 U M R Page 13 containing 3, 3, 4 boys so that the oldest boy is in the group of 4 boys and youngest boy is in a group of three boys is 8 2 36 × . Statement – 2: The number of ways in which ( ) 2m n + objects can be divided in to three groups containing , , m m n ( ) m n ≠ objects is ( )2 2 2 m n m n + Key: A 30. Statement – 1: The number of ways of distributing n identical objects into r distinct boxes so that each box is non-empty is ( ) 1 1 n r r C + − − Statement – 2: The number of ways of arranging n objects of one kind and ( ) 1 r − objects of a different kind in a row in ( ) 1 1 r n r C − + − . Key: D 31. Statement – 1:Let { } , , , , A a b c d e = and : f A A → be an onto function such that ( ) f x x x A ≠ ∀ ∈ . Then the number of such functions f is 44. Statement – 2: The number of derangements of n objects is ( ) 1 1 1 1 ...... 2 3 4 n n n ⎛ ⎞ − − + + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Key: A 32. Statement – 1: The maximum number of points of intersection of 8 circles is 56. Statement – 2: The maximum number of points of intersection of 4 circles and 4 straight lines is 50. Key: B 33. Statement – 1: The maximum value of k Such that 50k divides 100 is 2. Statement – 2: If P is a prime then the maximum exponent of P contained in n is 2 3 ..... n n n p p p ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ + + + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Key: D 34. Statement – 1: There are 5 integers between 1 and 200 each of which has 10 divisors. Statement – 2:If 1 2 3 , , p p p are distinct primes then the number of positive integral 3 1 2 1 2 3 , , p p pα α α divisors of i N α∈ is ( )( )( ) 1 2 3 1 1 1 α α α + + + . Key: A 35. Statement – 1: The number of ways in which a set of 10boys can be divided into three groups containing 3, 3, 4 boys so that the oldest boy is in the group of 4 boys and youngest boy is in a group of three boys is 8 2 36 × . Statement – 2: The number of ways in which ( ) 2m n + objects can be divided in to three groups containing , , m m n ( ) m n ≠ objects is ( )2 2 2 m n m n + Key: A 36. Statement – 1: The number of ways of distributing n identical objects into r distinct boxes so that each box is non-empty is ( ) 1 1 n r r C + − − Statement – 2: The number of ways of arranging n objects of one kind and ( ) 1 r − objects of a different kind in a row in ( ) 1 1 r n r C − + − . Key: D 37. Statement – 1: If n is an odd prime then integral part of ( ) 1 2 5 2 n n+ + − is divisible by 20 n. Statement – 2:If n is prime 1 r n < < then n r C is divisible by n. Key: A 38. Statement – 1:Let { } , , , , A a b c d e = and : f A A → be an onto function such that ( ) f x x x A ≠ ∀ ∈ .Then the number of such functions f is 44. Statement – 2: The number of derangements of n objects is ( ) 1 1 1 1 ...... 2 3 4 n n n ⎛ ⎞ − − + + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Key: A U.M.R. Permutations and Combinations 14 U M R Page 14 SECTION – IV Linked Comprehension Type This section contains paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. Passage :I 5 letters are supposed to address for 5 people whose addresses are mentioned on envelopes. In how many ways letters are dearrange in envelopes. 1. All the 5 letters are dearranged (i.e all the letters are placed in wrongly addressed) in envelopes a) 120 b) 119 c) 44 d) None of these 2. At least three letters are placed in wrongly addressed envelopes a) 109 b) 110 c) 119 d) None of these 3. Exactly one letter is placed in wrongly addressed envelope a) 2 b) 1 c) 6 d) None of these Hint : Passage : II Let A be a set of n elements. Let 1 2 r p ,p ,.....p be r properties associated with one or more elements of A. Let Ai be the set of elements of A which satisfy at least the property pi;. Ai Aj ∩ be the set of elements of A which satisfy at least the properties Pi and Pj . Let sk be the number of elements of A which satisfy at least k of the properties. Then ( ) ( ) r 1 i 2 i j i 1 1 i J r s nA,s nA A = ≤< ≤ = = ∩ ∑ ∑∑ , etx. ( ) ( )r 1 1 2 r 1 2 3 n A A .... A s s s ..... 1 + ∪ ∪ = − + + − r s . 01. Out of first 250 natural numbers, the numbers which are divisible by 6 to 10 or 15 is a) 82 b) 74 c) 176 d) 66 02. Out of the first 250 natural numbers, the number of numbers which are neither divisible by 4 nor by 6 nor by 10 is a) 95 b) 4 c) 159 d) 155 03. Number of ways of selecting 6 cards out of 52 cards so that there is at least one card from each suit is n. Mark all the correct alternatives for n. a) 6 6 52C 4 39C − × b) ( ) ( ) ( ) 3 2 2 3 2 13C 13 6x 13C 13 × + c) ( ) 6 5 2 4 3 3 52C 4 39C 4C 26C 4C 13C − + − d) ( ) ( ) ( ) 52 39 26 13 6 6 6 6 C 4 C 6 C 4 C − + − Key : B Passage : III A is a set containing n elements . A subset S1 of A is chosen. The set A is reconstructed by replacing the element of S1. Again a sub set S2 of A is chosen and again the set is reconstructed by the replacing the elements of S2. The number of ways of choosing S1 or S2 where . 01. S1 and S2 have one element common is (A) n 1 3 − (B) n 1 n.3 − (C) n 1 2 − (D) n 02. 1 2 S S A = U is (A) n 3 (B) n n.3 (C) n 4 (D) n 1 4 − 03. S1 is a subset of S2 is (A) n 1 4 − (B) n 1 3 + (C) n 4 (D) n 3 KEY : B, A, D Passage : IV If there are n A′ s, nB′ s and n C′ s making 3n letters. Then the number of ways of selecting r letter out of them. 01. When the value of [ ] r 1,n ∈ is (A) r 1 C (B) r 1 2 C + (C) r 2 2 C + (D) None of these 02. When the value of [ ] r n 1,2n 1 ∈ + + is (A) r 2 r n 1 2 2 C 3. C + − + − (B) r r n 1 2 2 C 3. C + − − (C) r 1 r n 1 2 3 C 3. C + + − − (D) r r n 1 2 2 C 3. C + − − 03. When the value of [ ] 2 2,3 r n n ∈ + is (A) r 2 r n 1 r 2n 2 2 2 C 3. C C + − + − − + (B) r 2 r n 1 r 2 2 2 C 3. C 3C + − + − + (C) r 2 r n 1 r 2 2 2 C 3. C 3.C + + − − − (D) r 2 r n 1 r 2n 2 2 2 C 3. C 3. C + − + − − + Key : C,A,D Passage : V Let p be a prime number and n be a positive integer then exponent of p in ! n is denoted by ( ) ! P E n and is given by [ ] 2 3 ! .... p n n n n n E n p P P p ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + + + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ where 1 k k p n p+ < < and [x] denotes the integer part U.M.R. Permutations and Combinations 15 U M R Page 15 of x 01. The exponent of 7 in 100 50 c is a) 0 b) 1 c) 2 d) 3 02. The number of zeros at the end of 108! Is a) 10 b) 13 c) 25 d) 26 03. The last non zero digit in 20! must be equal to a) 2 b) 4 c) 6 d) 8 Key: A,C,B Passage : VI Consider all possible permutations of the word ENDEANOEL 01. The number of permutations containing the word ENDEA (a) 5 (b) 2 5 (c) 7 (d) 6 ! 02. The number of permutations in which the letter E occurs in the first and the last position is (a) 5 (b) 21 5 × (c) 2 5 × (d) 4 5 × 03. The number of permutations in which A, E, O occur in odd places only (a) 5 (b) 6 (c) 7 (d) 2 5 × Hint : a, b, d Passage : VII A Triangle is called an integer triangle if all the sides are Integers If a, b, c are sides of an Integer Triangle then we can assume that a b c ≤ ≤ (any other permutation will yield same triangle since sum of two sides is greater than the third side If c is fixed a + b will vary from c + 1 to 2c the number of such integer triangles can be formed by finding integer solutions if 1, 2...... 2 a b c a b c a b c + = + + = + + = 01. The number of integer isosceles or equilateral triangles none of whose sides exceed 4 must be (a) 9 (b) 10 (c) 11 (d) 12 02. The number of integer isosceles or equilateral triangles none of whose sides exceed 2c must be (a) 2 c (b) 2 2c (c) 2 3c (d) 2 32c 03. If c is fixed and odd the number of integer isosceles or equilateral triangles whose sides are a, b, c a b c ≤ ≤ must be (a) 2 1 2 c − (b) 2 1 2 c + (c) 3 1 2 c + (d) 3 1 2 c − Key : d, c, d Passage : VIII There are ‘n’ intermediate stations on a railway line from one terminus to another. In how many ways can the train stop at 3 of these intermediate stations, if 01. All the three stations are consecutive a) (n + 2) b) (n + 1) c) (n – 1) d) (n – 2) 02. Atleast two of the stations are consecutive a) (n + 2) (n -1) b) (n – 2) (n – 1) c) (n – 2)2 d) None 03. No two of these stations are consecutive a) 3 c n b) ( ) 3 2 c n − c) ( )( ) 2 3 6 n n − − d) none Key: D,C,B Passage : IX Consider the network of equally spaced parallel lines (6 horizontal and 9 vertical) shown in the figure. All small squares are of the same size. A shortest route from A to C is defined as a route consisting 8 horizontal steps and 5 vertical steps. Since any shortest route is a typical arrangement of 8H and 5V. The number of shortest route = 13! 5!8! . Answer the following questions : P Q R A BC D 01. The number of shortest routes through the junction P a) 240 b) 216 c) 560 d) None of these U.M.R. Permutations and Combinations 16 U M R Page 16 02. The number of shortest routes which go following street PQ must be a) 324 b) 350 c) 512 d) None of these 03. The number of shortest routes which pass through junctions P and R a) 144 b) 240 c) 216 d) None of these Hint: C,B,B Passage : IX The number of non negative integer solutions of the equation r1x1 + r2x2 + … + rnxn = m where x1, x2, … , xn are the variables and ri are either 1 or greater than 1. If ri = 1 for all i = 1, 2, …, n then it is known that the number of non negative integer solutions is m+n-1Cn-1 and number of positive integer solution is m-1Cn-1. If ri > 1 for some i then we can find the number of integer solutions by assigning all possible integer values to xi. Answer the following questions : 01. The number of integer solutions of the equation 3x1 + 5x2 + x3 + x4 = 10 must be a) 33 b) 34 c) 35 d) 36 02. The number of non negative integer solutions of 5x1 + x2 + x3 + x4 = 14 must be a) 15C2 + 10C2 + 5C2 b) 16C2 + 11C2 + 6C2 c) 16C3 + 11C3 + 6C3 d) None of these 03. The number of positive integer solutions to the equation(x1 + x2 + x3)(y1 + y2 + y3 + y4) = 77 must be a) 9 14 13 10 2 3 2 3 C C C C × + × b) ( )2 9 14 2 3 C C × c) 11 15 14 11 2 3 2 3 C C C C × + × d) None of these Key : D,D,B Passage-X Consider all permutations of the letters of the word MORADABAD 01. The no. of permutations which contain the word BAD is a) 21 5! × b) 7 5! × c) 6 5! × d) 2 5! × 02. The no. of permutations with the letter D occurring in the first and the last places is : a) 21 5! × b) 7 5! × c) 6 5! × d) 2 5! × 03. The no. of permutations with the letters M , A, O occurring only in odd positions is: a) 21 5! × b) 7 5! × c) 6 5! × d) 2 5! × Hint : A,B,D Passage-XI Given are six 0`s, five 1`s and four 2`s. consider all possible permutations of all these numbers. [ A permutation can have its leading digit 0]. 01. How many permutations have the first 0 preceeding the first 1? a) 15 10 4 5 C C × b) 15 10 5 4 C C × c) 15 10 6 5 C C × d) 15 10 5 5 C C × 02. In how many permutations does the first 0 preceed the first 1 and the first 1 preceed first 2. a) 14 8 5 6 C C × b) 14 8 5 4 C C × c) 14 8 6 4 C C × d) 14 8 6 6 C C × 03. The no. of permutations in which all 2`s are together but no two of the zeroes are together is a) 42 b) 40 c) 84 d) 80 Hint : A,B,A Passage:-XII There are 9 balls and 3 boxes, which can hold all the nine balls. A man wants to keep all these balls in the boxes and carry. 01. If the balls are of the same colour and identical and boxes are also identical. The no. of ways he can arrange the balls. a) 12 b) 55 c) ( )3 9! 3! d) ( )4 9! 3! 02. If the boxes are of different colour and balls are identical and of same colour. The no. of ways in which he can arrange the balls a) ( )3 9! 3! b) 55 c) 93 d) 165 03. If the balls are of different colour and the boxes are also of different colour and not identical. The no. of ways in which he can arrange the balls a) 531441 b) 729 c) 165 d) none Key: A, B, A U.M.R. Permutations and Combinations 17 U M R Page 17 Passage:-XIII Suppose we have to find all possible divisors of n = 21, 600, first of all we will find the exponent of all prime numbers occuring in it. So 21,600 = 25.33.52. Now the divisor will depend on the selection of prime-factors. If we take all possible selection of prime-factors, we will get all possible divisors. ∴No. of divisors = (5 + 1) (3 + 1) (2 + 1) = 72 01. The total no. of even divisors of 21,600 is (A) 60 (B) 30 (C) 50 D) 40 02. The total number of divisors of the form of 4m + 2(m ≥ 0) of 21,600 is (A) 60 (B) 72 (C) 12 (D) 50 03. The sum of all possible divisors of 21,600 is (A) 76120 (B) 78520 (C) 76240 (D) 78120 Key: C, B, D Passage:-XIV If there are n A′ s, nB′ s and n C′ s making 3n letters. Then the number of ways of selecting r letter out of them. 01. When the value of [ ] r 1,n ∈ is A) r 1 C B) r 1 2 C + C) r 2 2 C + D) None of these 02. When the value of [ ] r n 1,2n 1 ∈ + + is A) r 2 r n 1 2 2 C 3. C + − + − B) r r n 1 2 2 C 3. C + − − C) r 1 r n 1 2 3 C 3. C + + − − D) r r n 1 2 2 C 3. C + − − 03. When the value of [ ] 2 2,3 r n n ∈ + is A) r 2 r n 1 r 2n 2 2 2 C 3. C C + − + − − + B) r 2 r n 1 r 2 2 2 C 3. C 3C + − + − + C) r 2 r n 1 r 2 2 2 C 3. C 3.C + + − − − D) r 2 r n 1 r 2n 2 2 2 C 3. C 3. C + − + − − + Key: C, A, D Passage:XV Considering the rectangular hyperbola xy = 15!. The number of points (α, β) lying on it, where 01. α, β∈I, is A) 2016 B) 4032 C) 4033 D) 8064 02. α, β ∈I+ and HCF (α, β) = 1, is A) 64 B) 785 C) 4032 D) 94185 03. α, β∈I+ and α divides β, is A) 96 B) 511 C) 1344 D) 4032 Key: D, A, A Passage: XVI: 12 seats are to be occupied by 4 boys. 01. The number of possible arrangements if no two boys sit side by side. A) 4! 12 4 C × B) 5! 4! 3! × × C) 4! 9 4 C × D) 4! 8 4 C × 02. The number of possible arrangements if there should be at least two empty seats between any two boys A) 4! 6 4 C × B) (12 5 ) 3! 4 3 C C − × C) 12 4! 43! P × D) 8 4! 42! P × 03. The number of possible arrangements if each boy has exactly one neighbor A) 4! 9 2 C × B) 4! 9 5 C × C) 4! 8 3 P × D) 4! 9 2 P × Key: C, A, A Passage:XVII Let S be the set of the first 18 natural numbers. The number of ways of selecting from S. 01. Three numbers such that they are all consecutive or none of them are consecutive is A) 576 B) 600 C) 640 D) 680 02. Three numbers such that they form an AP is A) 60 B) 64 C) 72 D) 80 03. Two numbers such that the sum of their cubes is divisible by 3 is A) 21 B) 31 C) 45 D) 51 Key: A, C, D Passage: XVIII It can be observed that largest power of prime p contained in n! must be equal to ...... 3 2 + ⎥⎦ ⎤ ⎢⎣ ⎡ + ⎥⎦ ⎤ ⎢⎣ ⎡ + ⎥⎦ ⎤ ⎢⎣ ⎡ pn pn pn (where [x] denotes greatest integer ≤ x). The result intuitively follows since in the product n × × × × × .... 4 3 2 1 there are ⎥⎦ ⎤ ⎢⎣ ⎡ pn integers divisible by p. Among these ⎥⎦ ⎤ ⎢⎣ ⎡ pn integers there U.M.R. Permutations and Combinations 18 U M R Page 18 are ⎥⎦ ⎤ ⎢⎣ ⎡ 2 pn integers which are divisible by p2 and so on. If we isolate the power of each prime contained in any number N then N can be written as .... 7 . 5 3 2 4 3 2 1 α α α α = N where i α are non-negative integers. 01. The highest power of 3 contained in (50)! must be equal to A) 20 B) 21 C) 22 D) 23 02. If 50! is computed, the number of zeros at the end must be equal to A) 11 B) 12 C) 13 D) 15 03. n nC 2 will not be divisible by 4 if A) n is odd B) n is even but not a power of 2 C) n is divisible by 3 D) n is a power of 2. Key: C, B, D Passage :XIX In a competition, six teams A, B, C, D, E, F play each other in the preliminary rounds – called round robin tournament. Each game ends either in a win or loss . The winners is awarded two point while the loser awarded zero points . After the round robin tournament, the three teams with the highest scores move to the final rounds. It is observed that i)In the game between E and F team E won . ii)After each team had played four games, team A had 6 points, team B had 8 points and team C had 4 points iii)Team D, E and F had won their games against A, B and C respectively iv) Teams A, B and D had moved to the final round. 01. The difference between the scores of C and E after the round robin tournament is A) 2 B) 0 C) 4 D) 6 02. The winner of the games between C and E and C and A respectively are A) C, C B) C, A C) E , A D) E, C 03. The final standings after the round – robin tournament . (In case of a tie between two teams, the team winning the match between them is ranked first ) A) ABCDEF B) BADCEF C) BDACEF D) ABCEFD Key: B, B, C Passage: XX A palindrome is a finite sequence of characters that reads the same forward and backwards (ALAMABALA). Then 01. 7 digit palindromes, under the restriction that no digit may appear more than twice is A) 4 1 0 p B) 3 9 p C) 4 9 p D) 3 9 .9 p 02. The number of 8 digit palindromes under the same condition of the above problem is A) 4 8 p B) 3 9 p C) 4 9 .9 p D) 3 9.9p 03. Let 1 2 3 4 5 4 3 2 1 x x x x x x x x x be a nine digit palindrome such that either the sequence { } 1 2 3 4 5 , , , , x x x x x is a strictly ascending or strictly decending. Then the number of such palindromes is A) 4 9 .9 p B) 5 3 .9 p C) 5 9 .9 C D) 5 3 9C × Key: D, D, D Passage: XXI A fruit cum mathematician opens a crazy shop named a; b; c. Daily he gets n (of his choice) different fruits, each costing Rs 5 and of good quality. a is the number of ways of selecting r-1 fruits out of the n fruits, b is the number of ways of selecting r fruits and C is the number of ways of selecting r + 1 fruits. He announces that days ratio a : b : c . He sells r fruits to the first vendee guesses the number n and r. Then gives away smallest number of m fruits to the poor Children. He announces the remaining number of fruits n1 . The number of fruits r1 the next vendee buy at the rate of ¾ of the price provided the vended the decreasing ratio a1: b1 : c1. If a : b : c = 7: 6: 5 . Then 01. (n, r) = A) (79,71) B) (194 ,190) C) (142,77) D) (144, 44) 02. m = A) 15 B) 4 C) 3 D) 5 03. The correct guess for a1 : b1 : c1 is A) 5 : 4: 3 B) 7 : 5: 3 C) 9 : 7 : 5 D) 6 : 8 : 10 Key: C, C, A Passage: XXII There are 10 different pairs of shoes in a box. 01. The number of ways of selecting 4 shoes such that there is no pair is A) 3156 B) 3216 C) 3360 D) 3472 02. The number of ways of selecting 4 shoes such that there is exactly one pair is A) 1210 B) 1440 C) 1690 D) 1960 U.M.R. Permutations and Combinations 19 U M R Page 19 03. The number of ways of selecting 4 shoes such that there are two pairs is A) 24 B) 32 C) 44 D) 45 Key: C, B, D Passage: XXIII Consider the natural number N = 3600. 01. The sum of all the positive integral divisors of N is A) 12493 B) 21394 C) 34921 D) 41293 02. If the product of all the positive integral divisors of N is P and if 2x divides P, then the maximum value of x is A) 45 B) 90 C) 135 D) 180 03. If the sum of the reciprocals of all the positive integral divisors of N is R then [ ] R is (where [ ] R is the greatest integer less than or equal to R) A) 1 B) 2 C) 3 D) 4 Key: A, B, C Passage:XXIV Let A be a non –empty set. Then a binary operation on A is defined as a function from A A × to A. If * is a binary operation defined on A then the image of ( ) , a b A A ∈ × is denoted by * a b.A binary operation * defined on A is said to be communicative if * * , a b b a a b A = ∀ ∈ Suppose { } ( ) . (, ) | , A n A A a b a b A = × = ∈ . 01. The number of binary operations that can be defined on A is A) n n B) 2n C) 2 n n D) 2 2n 02. If 1 n > then the number of binary operations that can be defined on A which are one-one is A) n B) 2 n C) 2 n n P D) 0 03. The number of commutative binary operations that can be defined on A is A) ( ) 1 n n n − B) ( ) 1 2 n n n − C) ( ) 1 n n n + D) ( ) 1 2 n n n + Key: C,D,D Passage:XXV For a finite set A, Let A denote the number of elements in the set A. Also let F denote the set of all functions { } { }( ) : 1, 2,..., 1, 2,...., 3, 2 f n kn k → ≥ ≥ satisfying ( ) ( ) 1 f i f i ≠ + for every i , 1 1 i n ≤ ≤ − . 01. F = A) ( ) 1 n k k− B) ( ) 1 n k k− C) ( ) 1 1 n k k − − D) ( ) 1 1 n k k − − 02. If ( ) , n k c denote the number of functions in F satisfying ( ) ( ) 1 , f n f ≠ then, for 4 n ≥ ( ) , , n k c = A) ( ) ( ) 1 1 1, n k k n k − − − − c B) ( ) ( ) 1 1, 1 n k k n k − − − − c C) ( ) ( ) 1 1 1, n k k n k − − − − c D) ( ) ( ) 1 1, n k k n k − − − c 03. For , ( , ) n k nk ≥ c , where ( ) , n k c has the same meaning as in Q.02, equals A) ( ) ( ) 1 1 n nk k + − − B)( ) ( ) ( ) 1 1 1 1 n n k k − − + − − C) ( ) ( ) ( ) 1 1 1 n n k k − + − − D) ( ) ( ) 1 1 1 n nk k − + − − Key: D,A,C Passage:XXVI A is a set containing n elements. A subset S1 of A is chosen. The set A is reconstructed by replacing the elements of S1. Again, a subset S2 of A is chosen and again the set is reconstructed by replacing the elements of S2. The number of ways of choosing S1 or S2 where 01. S1 and S2 have one element common is A)3n–1 B)n . 3n–1 C)2n–1 D)n 02. S1 ∪ S2 = A is A)3n B)n . 3n C)4n D)4n–1 03. S1 is a subset of S2 is A)4n–1 B)3n + 1 C)4n D)3n Key: B,A,D SECTION – IV Matrix Match Type This section contains questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in Column I have to be matched with statements (p,q,r,s) in Column II. U.M.R. Permutations and Combinations 20 U M R Page 20 01. Match the following: Column -I Column-II A) Rank of TOSS if the letters of the p) 216 word are arranged in dictionary order B)The rank of COCHIN of the letters of q) 97 the word are arranged in dictionary order C) How many different NINE digited numbers can be r) 10 formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions D) A five digited number divisible by 3 is to be formed s) 60 using the numbers 0.1,2,3,4 & 5 without repetitions. The total number of ways this can be done are 02. Match the following Column-I Column-II A. When three Dice are rolled the number p. 54 of possible out comes in which at least one die shows 6 is B. The number of even proper divisors of 1008 is q. 56 C. The number of ways of selecting 10 balls from unlimited r. 23 red, green, white and yellow balls,if selection must include atleast 2 red and 3 yellow balls is …….. ( where balls of the same colour are alike D. The number of ve + integral solutions of the s. 76 equation 140 xyz = is …. Key : A –s, B – s, C – q, D – p 03. Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II. Column – I Column – II A) The number of permutations p) 5! containing the word ENDEA, is B) The number of permutations in which the q) 2 5! × letter E occurs in the first and the last positions, is C) The number of permutations in which none of r) 7 5! × the letters D, L, N occur in the last five positions, is D) The number of permutations in which the letters s) 21 5! × A, E, O occur only in odd positions, is Key: A p;B s;C q;D q → → → → 04. Match the following: Column-I Column-II A) The number of ways of answering one or more of n different questions is p) 2n r Pr B) The number of ways of answering one or more of n different questions q) 2n when each question has an alternative is C) The number of circular permutations of n different things taken r at a time is r) n r Pr D) The number of circular permutations of n different things taken r at a time, s) 3 1 n − given that an anticlockwise and a clockwise arrangement in the same order are considered to be equivalent is t) 2 1 n − Key : A-t, B-s, C-r, D-p 05. Match the following: Given a convex octagon. The no. of triangles that can be formed having Column-I Column-II A) one side common with the octagon p) 16 B) two sides common with the octagon q) 7 C) no side common with the octagon r) 32 D) the number of diagonals of the octagon s) 20 t) 8 Key : A-r, B-t, C-p, D-s 06. Match the values given in Column II with the quantities given in Column I Column-I: Column-II: U.M.R. Permutations and Combinations 21 U M R Page 21 A) Let abc = 8 (a, b, c 1) matches during his career and made a total of ( )( ) 1 1 2 2 4n n n + + − − runs. If the player made 1 .2n k k − + runs in the k th match ( ) 1 k n ≤ ≤ ,find n. Key: 7 25. Let X = {1, 2, 3, ... 100} and Y be a subset of X such that the sum of no two elements in Y is divisible by 7. If the maximum possible number of element in Y is 40 + λ then λ is Key: 5 26. Let X = {1, 2, 3, ... 100} and Y be a subset of X such that the sum of no two elements in Y is divisible by 7. If the maximum possible number of element in Y is 40 + λ then λ is KEY : 5 27. There four balls of different colours and four different boxes in size but colours same as those of balls. The number of ways in which the balls, one each in a box, could be found such that a ball doesn’t go to a box of its own colour is ---Key: 9