Class 11 Mathematics Permutations and Combinations MCQs Set 06

PERMUTATIONS OF DISSIMILAR THINGS

Question. The number of ways in which 5 Boys and 5 Girls can be arranged in a row so that no two girls are together is
(a) 10!
(b) \( 5! \cdot 6! \)
(c) \( (5!)^{2} \)
(d) \( 2(5!)^{2} \)
Answer: (b) \( 5! \cdot 6! \)

Question. The number of ways can 4 men, 3 boys, 2 women be seated in a row so that the men, the boys and the women are not seperated is
(a) \( 4! \cdot 3! \cdot 2! \)
(b) \( (4!)^{2} \cdot 3! \cdot 2! \)
(c) \( 4! \cdot (3!)^{2} \cdot 2! \)
(d) \( 4! \cdot 3! \cdot (2!)^{2} \)
Answer: (c) \( 4! \cdot (3!)^{2} \cdot 2! \)

Question. A, B, C are three persons among 7 persons who speak at a function. The number of ways in which it can be done if 'A' speaks before 'B' and 'B' speaks before 'C' is
(a) 820
(b) 830
(c) 840
(d) 850
Answer: (c) 840

Question. The number of ways in which 20 different white balls and 19 different black balls be arranged in a row, so that no two balls of the same colour come together is
(a) \( 20! \cdot ^{21}P_{19} \)
(b) \( 20! \cdot 19! \)
(c) \( (20!)^{2} \)
(d) \( (21)! \cdot ^{20}C_{19} \)
Answer: (b) \( 20! \cdot 19! \)

Question. There are 10 white and 10 black balls marked 1,2,3 ..... 10. The number of ways in which we can arrange these balls in a row in such a way that neighbouring balls are of different colours is
(a) \( 10! \cdot 9! \)
(b) \( 20! \)
(c) \( (10!)^{2} \)
(d) \( 2(10!)^{2} \)
Answer: (d) \( 2(10!)^{2} \)

Question. The number of ways in which 10 books can be arranged in a row such that two specified books are side by side is
(a) \( \frac{10!}{2!} \)
(b) 9!
(c) \( 9! \cdot 2! \)
(d) \( \frac{9!}{2!} \)
Answer: (c) \( 9! \cdot 2! \)

Question. The number of ways in which the time table for Monday be completed if there must be 5 lessons that day (Algebra, Geometry, Calculus, Trigonometry, Vectors) and Algebra and Geometry must not immediately follow each other are
(a) 72
(b) 5!
(c) \( 3 \times 5!/2 \)
(d) 6!
Answer: (a) 72

Question. If 'a' denotes the number of permutations of \( x + 2 \) things taken all at a time, b the number of permutation of x things taken 11 at a time and c the number of permutations of \( x - 11 \) things taken all at a time such that \( a = 182bc \), then the value of x is
(a) 15
(b) 12
(c) 10
(d) 18
Answer: (b) 12

Question. The number of four digit numbers that can be formed with 0, 1, 2, 3, 4, 5 is
(a) \( ^{6}P_{4} \)
(b) \( 5 \cdot ^{6}P_{3} \)
(c) \( ^{6}P_{4} - ^{5}P_{4} \)
(d) \( ^{6}P_{4} - ^{5}P_{3} \)
Answer: (d) \( ^{6}P_{4} - ^{5}P_{3} \)

Question. The number of four digit odd numbers that can be formed with 1, 2, 3, 4, 5, 6, 7, 8, 9 is
(a) \( 4 \cdot ^{8}P_{3} \)
(b) \( 5 \cdot ^{8}P_{3} \)
(c) \( 4 \cdot ^{7}P_{3} \)
(d) \( 5 \cdot ^{7}P_{3} \)
Answer: (b) \( 5 \cdot ^{8}P_{3} \)

Question. The number of four digit odd numbers that can be formed so that no digit being repeated in any number is
(a) 2240
(b) 2420
(c) 2440
(d) 2520
Answer: (a) 2240

Question. The number of four digit even numbers that can be formed with 0, 1, 2, 3, 7, 8 is
(a) 180
(b) 175
(c) 160
(d) 156
Answer: (d) 156

Question. The number of numbers of 9 different non-zero digits such that all the digits in the first four places are less than the digit in the middle and all the digits in the last four places are greater than the digit in the middle is
(a) \( 2(4!) \)
(b) \( (4!)^{2} \)
(c) \( 8! \)
(d) \( 2 \times (4!)^{2} \)
Answer: (b) \( (4!)^{2} \)

Question. Number of 6-digit telephone numbers, which can be constructed with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if each number starts with 35 and no digit appears more than once is
(a) 1680
(b) 8!
(c) 6!
(d) \( 6 \cdot 6! \)
Answer: (a) 1680

PROBLEMS ON DIVISIBILITY

Question. If repetitions are not allowed, the number of numbers consisting of 4 digits and divisible by 5 and formed out of 0, 1, 2, 3, 4, 5, 6 is
(a) 220
(b) 240
(c) 370
(d) 588
Answer: (a) 220

Question. The number of seven digit numbers divisible by 9 formed with digits 1, 2, 3, 4, 5, 6, 7, 8, 9 without repetition is
(a) 7!
(b) \( ^{9}P_{7} \)
(c) \( 3(7!) \)
(d) \( 4(7!) \)
Answer: (d) \( 4(7!) \)

Question. A five-digit number divisible by 6 is to be formed by using 0, 1, 2, 3, 4, 5 without repetition. The number of ways in which this can be done is
(a) 60
(b) 48
(c) 108
(d) 216
Answer: (c) 108

Question. A 5 digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4, 5 without repetition. The total number of ways this can be done is
(a) 120
(b) 96
(c) 216
(d) 220
Answer: (c) 216

Question. The number of 5 digited numbers which are not divisible by 5 and which contains of 5 odd digits is
(a) 96
(b) 120
(c) 24
(d) 32
Answer: (a) 96

Question. The sum of integers from 1 to 100 that are divisible by 2 or 5 is
(a) 3000
(b) 3050
(c) 3600
(d) 3250
Answer: (b) 3050

Question. The number of five digit numbers that can be formed with 0, 1, 2, 3, 5 which are divisible by 25 is
(a) 42
(b) 24
(c) 10
(d) 38
Answer: (c) 10

Question. The number of ways in which we can arrange the digits 1, 2, 3....9 such that the product of five digits at any of the five consecutive positions is divisible by 7 is
(a) 7!
(b) \( ^{9}P_{7} \)
(c) 8!
(d) \( 5(7!) \)
Answer: (c) 8!

SUM OF THE NUMBERS

Question. The sum of the digits at the ten's place of all the numbers formed with the digits of 3, 4, 5, 6 taken all at a time is
(a) 432
(b) 108
(c) 36
(d) 18
Answer: (b) 108

Question. The sum of the value of the digits at the ten's place of all the numbers formed with the help of 3, 4, 5, 6 taken all at a time is
(a) 1080
(b) 4320
(c) 360
(d) 180
Answer: (a) 1080

Question. The sum of all the four digit numbers that can be formed with 0, 2, 3, 5 is
(a) 66660
(b) 66480
(c) 64440
(d) 65520
Answer: (c) 64440

PERMUTATIONS WHEN REPETITIONS ARE ALLOWED & LIKE THINGS

Question. The number of permutations that can be made out of the letters of the word "ENTRANCE" so that the two 'N's are always together is
(a) \( \frac{7!}{(2!)^{2}} \)
(b) \( 7! \)
(c) \( \frac{7!}{2!} \)
(d) \( \frac{7!}{(2!)^{3}} \)
Answer: (c) \( \frac{7!}{2!} \)

Question. The number of permutations that can be made by using all the letters of the word TATATEACUP that start with A and end with U is
(a) 5040
(b) \( 8! \cdot 5! \cdot 3! \cdot 3! \)
(c) 3360
(d) 360
Answer: (c) 3360

Question. All the letters of the word EAMCET are arranged in all possible ways. The number of such arrangements in which no two vowels are adjacent to each other is
(a) 36
(b) 54
(c) 72
(d) 144
Answer: (c) 72

Question. If the number of ways in which n different things can be distributed among n persons so that at least one person does not get any thing is 232. Then n is equal to
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (b) 4

Question. Number of ways of permuting the letters of the word "ENGINEERING" so that the order of the vowels is not changed is
(a) \( ^{11}P_{5} \)
(b) \( \frac{11!}{5!} \)
(c) \( \frac{11!}{2!5!} \)
(d) \( \frac{^{11}P_{5}}{2} \)
Answer: (d) \( \frac{^{11}P_{5}}{2} \)

Question. The number of different numbers that can be formed by using all the digits 1, 2, 3, 4, 3, 2, 1 so that odd digits always occupy the odd places is
(a) 6
(b) 72
(c) 60
(d) 18
Answer: (d) 18

Question. A three digit number n is such that the last two digits of it are equal and different from the first. The number of such n's is
(a) 64
(b) 72
(c) 81
(d) 900
Answer: (c) 81

Question. The number of three digit numbers having only two consecutive digits identical is
(a) 153
(b) 162
(c) 168
(d) 163
Answer: (b) 162

Question. The number of 'n' digit numbers such that no two consecutive digits are same is
(a) 9!
(b) \( 9^{n} \)
(c) \( 9^{n} \)
(d) 9n
Answer: (c) \( 9^{n} \)

Question. The number of five digit numbers formed using the digits 0, 2, 2, 4, 4, 5 which are greater than 40,000 is
(a) 84
(b) 90
(c) 72
(d) 60
Answer: (b) 90

Question. The number of numbers greater than or equal to 1000 but less than 4000 that can be formed with 0, 1, 2, 3, 4 so that any digit may be repeated is
(a) 374
(b) 375
(c) 120
(d) 360
Answer: (b) 375

Question. The sum of all 3 digited numbers that can be formed from the digits 1 to 9 and when the middle digit is a perfect square is (repetitions are allowed)
(a) 1,34,055
(b) 2,70,540
(c) 1,70,055
(d) 2,34,520
Answer: (a) 1,34,055

PROBLEMS ON NUMBER OF FUNCTIONS

Question. Number of functions from Set-A containing 5 elements to a set-B containing 4 elements is
(a) \( 5^{4} \)
(b) \( 4^{5} \)
(c) \( 4! \)
(d) \( 5! \)
Answer: (b) \( 4^{5} \)

Question. The number of one one functions that can be defined from \( A = \{a, b, c\} \) into \( B = \{1, 2, 3, 4, 5\} \) is
(a) \( ^{5}P_{3} \)
(b) \( ^{5}C_{3} \)
(c) \( 5^{3} \)
(d) \( 3^{5} \)
Answer: (a) \( ^{5}P_{3} \)

Question. The number of many one functions from \( A = \{1, 2, 3\} \) to \( B = \{a, b, c, d\} \) is
(a) 64
(b) 24
(c) 40
(d) 0
Answer: (c) 40