Class 11 Mathematics Permutations and Combinations MCQs Set 10

PERMUTATIONS OF DISSIMILAR THINGS

Question. The letters of the word ‘HEXAGON’ are arranged in all possible ways. If the order of the vowels is not to be considered then the number of possible arrangements is
(a) 1680
(b) 840
(c) 420
(d) \( 8!/2! \)
Answer: (b) 840

Question. The letters of the word ‘MADHURI’ are arranged in all possible ways. The number of arrangements in which there are 2 letters between R and D is
(a) 360
(b) 480
(c) 960
(d) 720
Answer: (c) 960

Question. The number of ways of arranging 8 Eamcet Question papers so that best and worst never come together is
(a) 30240
(b) 21600
(c) 5040
(d) 4320
Answer: (a) 30240

Question. The number of even numbers between 200 and 20,000 can be formed with the digits 0, 1, 2, 3, 4 when repetitions are not allowed is
(a) 144
(b) 120
(c) 99
(d) 96
Answer: (c) 99

Question. The number of ways to arrange the letters of the word ‘GARDEN’ with vowels in alphabetical order is
(a) 360
(b) 240
(c) 120
(d) 480
Answer: (a) 360

SUM OF THE NUMBERS

Question. The total number of seven-digit numbers such that the sum of whose digits is even is
(a) \( 9 \times 10^6 \)
(b) \( 45 \times 10^5 \)
(c) \( 81 \times 10^5 \)
(d) \( 9 \times 10^5 \)
Answer: (b) \( 45 \times 10^5 \)

Question. The sum of all the numbers that can be formed by taking all digits 2, 3, 4, 4, 5 only is
(a) 2399976
(b) 2339976
(c) 2333976
(d) 2399376
Answer: (a) 2399976

PROBLEMS BASED ON REPETITIONS & SIMILAR THINGS

Question. The number of ways in which six ‘+’ and four ‘-’ signs can be arranged in a line such that no two ‘-’ signs come together is
(a) 35
(b) 120
(c) 720
(d) 610
Answer: (a) 35

Question. The number of ways in which the letters of the word "SUCCESSFUL" be arranged such that the 'S's and 'U's will come together is
(a) \( 7! \)
(b) \( \frac{7!}{2!} \)
(c) \( \frac{7!}{2!2!} \)
(d) \( \frac{7!}{2! 2! 2!} \)
Answer: (b) \( \frac{7!}{2!} \)

Question. The number of ways in which the letters of the word "SUCCESSFUL" be arranged such that No two 'S's will come together is
(a) \( \frac{7!}{2!2!} \)
(b) \( \frac{7!}{2!2!} \cdot ^8P_3 \)
(c) \( ^8P_3 \)
(d) \( \frac{7!}{2! 2! 3!} \cdot ^8P_3 \)
Answer: (d) \( \frac{7!}{2! 2! 3!} \cdot ^8P_3 \)

Question. The number of 6 digited numbers less than 4,00,000 can be formed by using the digits 1, 2, 3, 3, 3, 4 is
(a) 150
(b) 100
(c) 50
(d) 200
Answer: (b) 100

Question. A library has 6 copies of one book, 4 copies of each of two books, 6 copies of each of three books and single copies of 8 books. The number of arrangements of all the books in a linear shelf is
(a) \( \frac{40!}{(2!)^4 (3!)^6} \)
(b) \( \frac{40!}{6! \cdot (4!)^2 (6!)^3} \)
(c) \( \frac{40!}{6! \cdot 4! \cdot 6!} \)
(d) \( \frac{40!}{4! \cdot (4!)^3 \cdot 6!} \)
Answer: (b) \( \frac{40!}{6! \cdot (4!)^2 (6!)^3} \)

Question. The number of different numbers each of six digits that can be formed by using the digits of the numbers 2,2,3,3,9,9 is
(a) 90
(b) 100
(c) 600
(d) 120
Answer: (a) 90

DEARRANGEMENT PROBLEMS

Question. There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that every ball does not go to a box of its own colour is
(a) 9
(b) 13
(c) 8
(d) 4
Answer: (a) 9

Question. There are seven greeting cards, each of a different colour and seven envelopes of the same seven colours. The number of ways in which the cards can be put in the envelopes so that exactly four of the cards go into the envelopes of the right colours is
(a) \( 2 \times ^7C_3 \)
(b) \( ^7C_3 \)
(c) \( 3! \times ^4C_3 \)
(d) \( 3! \times ^7C_3 \times ^4C_3 \)
Answer: (a) \( 2 \times ^7C_3 \)

CIRCULAR PERMUTATIONS

Question. The number of ways in which 7 Indians and 6 Pakistanies sit around a round table so that no two Indians are together is
(a) \( (7!)^2 \)
(b) \( (6!)^2 \)
(c) \( 6! \ 7! \)
(d) zero
Answer: (d) zero

Question. Number of ways in which 7 seats around a table can be occupied by 15 persons is
(a) \( ^{15}P_7 \)
(b) \( ^{15}C_7 / 7 \)
(c) \( ^{15}P_7 / 7 \)
(d) \( 14! \)
Answer: (c) \( ^{15}P_7 / 7 \)

OTHER MODELS

Question. A binary sequence is an array of 0’s and 1’s. The number of n-digit binary sequences which contain even number of 0’s is
(a) \( 2^{n-1} \)
(b) \( 2^n - 1 \)
(c) \( 2^{n-1} - 1 \)
(d) \( 2^n \)
Answer: (a) \( 2^{n-1} \)

Question. The last digit of \( (1! + 2! + 3! + ......+2005!)^{500} \) is
(a) 9
(b) 2
(c) 7
(d) 1
Answer: (d) 1

Question. The number of positive terms in the sequence \( \{x_n\} \) where \( x_n = \frac{195}{4(n!)} - \frac{^{n+3}P_3}{(n+1)!} \) are
(a) 10
(b) 5
(c) 6
(d) 4
Answer: (d) 4

Question. The range of the function \( f(x) = {^{7-x}P_{x-3}} \) is
(a) {1, 2, 3}
(b) {1, 2, 3, 4}
(c) {1, 2, 3, 4, 5}
(d) {1, 2, 3, 4, 5, 6}
Answer: (a) {1, 2, 3}

Question. When \( n \) is a positive integer, then \( (n^2)! \) is
(a) divisible by \( n^2 \), but not by \( n! \)
(b) divisible by \( n! \), but not by \( (n!)^2 \)
(c) divisible by \( (n!)^2 \), but not by \( (n!)^n \)
(d) divisible by \( (n!)^n \)
Answer: (d) divisible by \( (n!)^n \)

Question. The number of distinct rational numbers p/q, where \( p, q \in \{1, 2, 3, 4, 5, 6\} \) is
(a) 23
(b) 32
(c) 36
(d) 63
Answer: (a) 23

Question. If m = number of distinct rational numbers p/q \( \in (0, 1) \) such that \( p, q \in \{1, 2, 3, 4, 5\} \) and n = number of onto mappings from {1, 2, 3} onto {1, 2}, then m - n is
(a) 1
(b) -1
(c) 0
(d) 3
Answer: (d) 3

Question. How many numbers lying between 100 and 10,000 can be formed with the digits 1, 2, 3, 4, 5 if atleast one digit in the same numbers are repeated
(a) \( 750 - {^5P_3} - {^5P_4} \)
(b) \( 750 - {^5P_3} + {^5P_4} \)
(c) \( 750 + {^5P_3} - {^5P_4} \)
(d) \( 750 + {^5P_3} + {^5P_4} \)
Answer: (a) \( 750 - {^5P_3} - {^5P_4} \)

Question. If all permutations of the letters of the word "AGAIN" are arranged as in dictionary, then fiftieth word is
(a) NAAGI
(b) NAGAI
(c) NAAIG
(d) NAIAG
Answer: (c) NAAIG

Question. The number of permutaions of the letters of the word HINDUSTAN such that neither the pattern ‘HIN’ nor ‘DUS’ nor ‘TAN’ appears, are :
(a) 166674
(b) 169194
(c) 166680
(d) 181434
Answer: (b) 169194

Question. The tens' digit of \( 1! + 2! + 3! + \dots + 29! \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. 5 students (A, B, C, D, E) are to be arranged in a row so, that A occupies the 2nd position and B is always adjacent to ‘C’, the number of such arrangements is
(a) 6
(b) 4
(c) 8
(d) 2
Answer: (c) 8

Question. No of different matrices that can be formed with elements 0, 1, 2, or 3 each matrix having 4 elements is
(a) \( 3 \times 2^4 \)
(b) \( 2 \times 4^4 \)
(c) \( 3 \times 4^4 \)
(d) \( 4^4 \)
Answer: (c) \( 3 \times 4^4 \)

Question. Sum of four digit numbers formed with 2, 3, 4, 5, using each digit any number of times is
(a) \( 4 \times 1111 \times 64 \)
(b) \( 14 \times 1111 \times 64 \)
(c) \( 14 \times 1111 \times 16 \)
(d) \( 4 \times 1111 \times 16 \)
Answer: (b) \( 14 \times 1111 \times 64 \)

Question. Total number of numbers that are less than \( 3 \times 10^8 \) and can be formed using the digits 1, 2, 3 is equal to
(a) \( \frac{1}{2}(3^9 + 4 \cdot 3^8) \)
(b) \( \frac{1}{2}(3^9 - 3) \)
(c) \( \frac{1}{2}(7 \cdot 3^8 - 3) \)
(d) \( \frac{1}{2}(3^9 - 3 + 3^8) \)
Answer: (c) \( \frac{1}{2}(7 \cdot 3^8 - 3) \)

Question. The number of numbers between 1 and \( 10^{10} \) which contain the digit 1 is :
(a) \( 10^{10} - 9^{10} + 1 \)
(b) \( 10^{10} - 9^{10} - 1 \)
(c) \( 10^{10} - 8^{10} + 1 \)
(d) \( 10^{10} - 9^{10} + 2 \)
Answer: (b) \( 10^{10} - 9^{10} - 1 \)

Question. No of symmetric matrices of order \( 3 \times 3 \) by using the elements of the set \( A = \{-3, -2, -1, 0, 1, 2, 3\} \) is
(a) \( 7^9 \)
(b) \( 7^6 \)
(c) \( 7^3 \)
(d) \( 7^9 - 7^3 \)
Answer: (b) \( 7^6 \)

Question. In the above problem No. of symmetric (or) skew symmetric matrix is
(a) \( 7^6 + 7^3 - 1 \)
(b) \( 7^9 + 7^6 \)
(c) \( 7^9 - 7^6 \)
(d) \( 7^9 - 7^6 - 1 \)
Answer: (a) \( 7^6 + 7^3 - 1 \)

Question. In a particular programming language a valid variable name can consist of a sequence of one to five alphanumeric characters A, B, C, D, ......, Z, 0, 1, 2, ....., 9 beginning with a letter. The number of valid variable name is
(a) \( \frac{26}{25}(26^5 - 1) \)
(b) \( \frac{26}{25}(26^5 + 1) \)
(c) \( \frac{26}{35}(36^5 + 1) \)
(d) \( \frac{26}{35}(36^5 - 1) \)
Answer: (d) \( \frac{26}{35}(36^5 - 1) \)

Question. If N is the sum of all digits used in writing all the numbers from 1 to 1000, then the last digit of N is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Consider 10 digits numbers of the form \( M = \sum_{r=0}^{9} a_r 10^r \), where \( a_r, r = 0, 1, 2, \dots, 9 \) are distinct digits, \( a_9 \neq 0 \), which are divisible by 99999. If N is the number of such numbers, then the last digit of N is
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (a) 6

Question. The number of arrangements of the letters of the word FORTUNE such that the order of vowels is unaltered is x, the order of consonents is unaltered is y, the order of vowels and consonents is unaltered is z, then \( \frac{x + y}{z} \) equals
(a) 10
(b) 20
(c) 30
(d) 40
Answer: (c) 30

Question. A variable name with atmost two characters in certain computer language must be either a alphabet or a alphabet followed by using the digit. Total number of different variable names that can exist in that language is equal to:
(a) 280
(b) 290
(c) 286
(d) 296
Answer: (c) 286