Permutations & Combinations PYQs Set 3 (PDF)

Set 3 of 11 · Questions 61-90 of 315

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Permutations & Combinations · Set 3 (Q61-90)
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61. $The least value of n so that^{(n - 1)} C_{3} +^{(n - 1)} C_{4} >^{n} C_{3}$ (1) $11$ (2) $9$ (3) $8$ (4) $7$ 62. $If the number of diagonals of a regular polygon is 35 , then the number of sides of the polygon is$ (1) $12$ (2) $9$ (3) $10$ (4) $11$ 63. $All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is$ (1) $327$ (2) $328$ (3) $324$ (4) $326$ 64. $If 5 dice are rolled simultaneously, then the number of ways of getting a total of seven of the numbers appear on their faces is$ (1) $12$ (2) $15$ (3) $20$ (4) $25$ 65. $Consider three boxes, each containing 10 balls labelled 1, 2, \dots, 10 Suppose one ball is randomly drawn from each of the boxes. Denote by n_{i}, the label of the ball drawn from the i^{th} box, (i = 1, 2, 3) . Then, the number of ways in which the balls can be chosen such that n_{1} < n_{2} < n_{3} is:$ (1) $120$ (2) $82$ (3) $240$ (4) $164$ 66. $In a shop there are five types of ice-creams available.. A child buys six ice-creams. Statement-1 : The number of different ways the child can buy the six ice-creams, is 10C5. Statement-2 : The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6A' s and 4B' s in a row.$ (1) $Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.$ (2) $Statement-1 is True,Statement-2 is True;Statement-2 is NOT a correct explanation for Statement-1$ (3) $Statement-1 is True, Statement-2 is False$ (4) $Statement-1 is False, Statement-2 is True$ 67. $From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:$ (1) $5148$ (2) $6084$ (3) $4356$ (4) $14950$ 68. $The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is$ (1) $36$ (2) $48$ (3) $60$ (4) $72$ 69. $Let S be the sample space of all five - digit numbers. If p is the probability that a randomly selected number from S, is a multiple of 7 but not divisible by 5, then 9p is equal to$ (1) $1.0146$ (2) $1.2085$ (3) $1.0285$ (4) $1.1521$ 70. $The sum of all 4-digit numbers that can be formed using the digits 2, 3, 4, 5, 6 without repetition, is$ (1) $533820$ (2) $532280$ (3) $533280$ (4) $532380$ 71. $The largest value of n, for which 40^{n} divides 60!, is$ (1) $11$ (2) $12$ (3) $14$ (4) $13$ 72. $Let P be the set of seven - digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set P is:$ (1) $173$ (2) $164$ (3) $158$ (4) $161$ 73. $All possible 5 digit numbers each having 5 distinct digits are formed using digits 1, 2, 3, 5, 6, 8 . Among them, the number of numbers which are divisible by 3 but not by 6 is$ (1) $120$ (2) $72$ (3) $48$ (4) $240$ 74. $The value of^{6} P_{4} + 4 \cdot^{6} P_{3} is$ (1) $5040$ (2) $2520$ (3) $840$ (4) $720$ 75. $If 2^{n} divides 16! and 2^{n + 1} does not divide 16!, then n =$ (1) $14$ (2) $15$ (3) $16$ (4) $17$ 76. $In an examination hall there are ' mn' chairs in m rows and n columns. The number of ways in which m students can be seated such that no row is vacant is$ (1) $m^{n} n!$ (2) $n^{m} m!$ (3) $m^{m} n!$ (4) $n^{n} m!$ 77. $The number of different ways of preparing a garland using 6 distinct white roses and 5 distinct red roses such that no two red roses come together is$ (1) $21600$ (2) $43200$ (3) $86400$ (4) $151200$ 78. $Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that Y \subseteq X, Z \subseteq X and Y \cap Z is empty, is :$ (1) $5^{2}$ (2) $3^{5}$ (3) $2^{5}$ (4) $5^{3}$ 79. $The number of 4 letter words that can be formed with the letters in the word EQUATION with at least one letter repeated is$ (1) $2400$ (2) $2408$ (3) $2416$ (4) $2432$ 80. $A student has to answer 10 questions, choosing at least 4 from each of the parts A and B . If there are 6 questions in part A and 7 in part B, then the number of ways can the student choose 10 questions is$ (1) $256$ (2) $352$ (3) $266$ (4) $426$ 81. $The number of ways of selecting two numbers a and b, a \in {2, 4, 6, \dots, 100} and b \in {1, 3, 5, \dots, 99} such that 2 is the remainder when a + b is divided by 23 is$ (1) $186$ (2) $84$ (3) $108$ (4) $268$ 82. $The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is$ (1) $720$ (2) $126 (5!)^{2}$ (3) $7 (360)^{2}$ (4) $7(720)$ 83. $There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____$ 84. $The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15, is$ 85. $Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is________$ 86. $Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, \frac{y}{9 x} =$ 87. $The total number of three-digit numbers, with one digit repeated exactly two times, is$ 88. $The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to____$ 89. $There are ten boys B1, B2, ...., B_{10} and five girls G_{1}, G_{2}, ...., G_{5} in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B1 and B2 together should not be the members of a group, is__________.$ 90. $The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is_____.$