Class 11 Mathematics Permutations and Combinations MCQs Set 09

PROBLEMS BASED ON \( ^n\text{P}_r \)

Question. 9 articles are to be placed in 9 boxes one in each box 4 of them are too big for three of the boxes. The number of possible arrangements is
(a) 9!
(b) 5! 4!
(c) \( ^6\text{P}_4 \times 5! \)
(d) 5! 6!
Answer: (c) \( ^6\text{P}_4 \times 5! \)

Question. The number of different ways that three distinct rings can be worn to 4 fingers with atmost one ring in each of the fingers is
(a) \( ^4\text{P}_3 \)
(b) \( ^5\text{P}_4 \)
(c) \( ^5\text{P}_3 \)
(d) \( ^4\text{P}_2 \)
Answer: (a) \( ^4\text{P}_3 \)

'\( n^r \)' MODEL

Question. There are 5 multiple choice questions in test. If the first three questions have 4 choices each and the next two have 5 choices each, the number of answers possible is
(a) 1500
(b) 1600
(c) 1700
(d) 1800
Answer: (b) 1600

Question. Sixteen men compete with one another in running, swimming and riding. How many prize lists could be made if there were altogether 6 prizes of different values, one for running, 2 for swimming and 3 for riding
(a) \( 16 \times 15 \times 14 \)
(b) \( 16^3 \times 15^2 \times 14 \)
(c) \( 16^3 \times 15 \times 14^2 \)
(d) \( 16^2 \times 15 \times 14 \)
Answer: (b) \( 16^3 \times 15^2 \times 14 \)

Question. The number of ways of wearing 6 different rings to 5 fingers is
(a) \( 5^6 \)
(b) \( 6^5 \)
(c) \( 5^5 \)
(d) \( 6^6 \)
Answer: (a) \( 5^6 \)

PERMUTATIONS OF DISSIMILAR THINGS

Question. The letters of the word "RANDOM" are arranged in all possible ways. The number of arrangements in which there are 2 letters between R and D is
(a) 36
(b) 48
(c) 144
(d) 72
Answer: (c) 144

Question. A family of 4 brothers and 3 sisters is to be arranged in a row for a photograph. The number of ways in which they can be seated if all the sisters are to sit together is
(a) 120
(b) 240
(c) 360
(d) 720
Answer: (d) 720

Question. If words are formed by taking only 4 at a time out of the letters of the word PHYSICAL, then the number of words in which Y occur is
(a) \( ^8\text{P}_4 \)
(b) \( \frac{7!}{3!} \)
(c) \( \frac{7!}{4!} \)
(d) \( ^7\text{C}_4 \)
Answer: (b) \( \frac{7!}{3!} \)

Question. The number of permutations alltogether of n things when r specified things are to be in an assigned order, though not necessarily consecutive is
(a) \( \frac{n!}{(n-r)!} \)
(b) \( \frac{n!}{(n-r)! r!} \)
(c) \( \frac{n!}{r!} \)
(d) \( (n-r)! \cdot r! \)
Answer: (c) \( \frac{n!}{r!} \)

Question. The number of ways of permuting the letters of the word DEVIL so that neither D is the first letter nor L is the last letter is
(a) 36
(b) 114
(c) 42
(d) 78
Answer: (d) 78

Question. The number of numbers lying between 10 and 1000 that can be formed with 0, 2, 3, 4, 5, 6 so that no digit being repeated in any number is
(a) 100
(b) 125
(c) 120
(d) 240

Answer: (b) 125

SUM OF THE NUMBERS

Question. The sum of all the numbers that can be formed by taking all the digits from 2, 3, 4, 5 is
(a) 93,324
(b) 79,992
(c) 66,66,600
(d) 78,456
Answer: (a) 93,324

Question. The sum of all 4 digited numbers that can be formed by taking the digits from 0, 1, 3, 5, 7, 9 is
(a) 16,23,300
(b) 16,32,300
(c) 16,32,030
(d) 16,33,200
Answer: (d) 16,33,200

Question. The sum of all 4 digited even numbers that can be formed from the digits 1, 2, 3, 4, 5 is
(a) 1,58,994
(b) 1,59,984
(c) 1,59,894
(d) 1,59,884
Answer: (b) 1,59,984

PROBLEMS BASED ON REPETITIONS & SIMILAR THINGS

Question. The number of numbers formed having not more than 6 digits using the numbers 3, 4, 5 when repetition is allowed is
(a) 1092
(b) 1090
(c) 1085
(d) 1064
Answer: (a) 1092

Question. The number of permutations of 25 dissimilar things taken more than 15 at a time when repetitions are allowed is (not exceeding 25)
(a) \( \frac{25}{24} \left(25^{25} - 25^{15}\right) \)
(b) \( \frac{25}{24} \left(25^{25} - 25^{10}\right) \)
(c) \( \frac{25}{24} \left(25^{25} + 25^{15}\right) \)
(d) \( \frac{25}{24} \left(25^{25} + 25^{10}\right) \)
Answer: (a) \( \frac{25}{24} \left(25^{25} - 25^{15}\right) \)

Question. A man has 3 servants. The number of ways in which he can send invitation cards to 6 of his friends through the servants is
(a) \( 3^6 \)
(b) \( 6^3 \)
(c) \( \frac{6!}{3!} \)
(d) \( ^6\text{P}_3 \)
Answer: (a) \( 3^6 \)

Question. In the word 'ENGINEERING' if all 'E's are not together and N's come together then number of permutations is
(a) \( \frac{9!}{2! 2!} - \frac{7!}{2! 2!} \)
(b) \( \frac{9!}{3! 2!} - \frac{7!}{2! 2!} \)
(c) \( \frac{9!}{3! 2! 2!} - \frac{7!}{2! 2! 2!} \)
(d) \( \frac{9!}{3! 2! 2!} - \frac{7!}{2! 2!} \)
Answer: (d) \( \frac{9!}{3! 2! 2!} - \frac{7!}{2! 2!} \)

Question. The number of permutations that can be made out of the letters of the word "MATHEMATICS" When all vowels come together is
(a) \( \frac{8! \cdot 4!}{2!} \)
(b) \( \frac{8! \cdot 4!}{(2!)^2} \)
(c) \( \frac{7! \cdot 4!}{2!} \)
(d) 7! 4!
Answer: (d) 7! 4!

Question. The number of permutation that can be made out of the letters of the word "MATHEMATICS". When no two vowels come together is
(a) \( 7! ^8\text{P}_4 \)
(b) \( \frac{7!}{2! 2!} \cdot ^8\text{P}_4 \)
(c) \( \frac{7! \cdot ^8\text{P}_4}{(2!)^3} \)
(d) \( 7! \frac{^8\text{P}_4}{2!} \)
Answer: (c) \( \frac{7! \cdot ^8\text{P}_4}{(2!)^3} \)

Question. The number of permutation that can be made out of the letters of the word "MATHEMATICS" When the relative positions of vowels and consonants remain unaltered is
(a) \( 3 \cdot 7! \)
(b) \( 2 \cdot 7! \)
(c) 7!
(d) \( 4 \cdot 7! \)
Answer: (a) \( 3 \cdot 7! \)

Question. The number of ways in which the letters of the word MULTIPLE be arranged without changing the order of the vowels is
(a) 3360
(b) 20160
(c) 6720
(d) 3359
Answer: (a) 3360

Question. If \( n \in N \) and \( 300 < n < 3000 \) and n is made of digits by taking from 0,1,2,3, 4,5 then greatest possible number of values of n is (repetition is allowed)
(a) 260
(b) 539
(c) 320
(d) 300
Answer: (b) 539

DEARRANGEMENT PROBLEMS

Question. \( f : A \to A, A = \{a_1, a_2, a_3, a_4, a_5\} \), then the number of one one functions so that \( f(x_i) \neq x_i, x_i \in A \) is
(a) 44
(b) 88
(c) 22
(d) 20
Answer: (a) 44

Question. The number of ways that all the letters of the word SWORD can be arranged such that no letter is in its original position is
(a) 44
(b) 32
(c) 28
(d) 20
Answer: (a) 44

Question. A person writes letters to six friends and addresses corresponding envelopes let x be the number of ways so that atleast two of the letters are in wrong envelopes and y be the number of ways so that all the letters are in wrong envelopes then \( x - y = \)
(a) 716
(b) 454
(c) 265
(d) 0
Answer: (b) 454

CIRCULAR PERMUTATIONS

Question. The no. of ways in which 6 gentlemen and 3 ladies be seated round a table so that every gentleman may have a lady by his side is ...
(a) 1440
(b) 720
(c) 240
(d) 480
Answer: (a) 1440

Question. The number of ways in which 7 men be seated at a round table so that two particular men are not side by side is
(a) 2400
(b) 120
(c) 360
(d) 480
Answer: (d) 480

Question. Six persons A, B, C, D, E and F are to be seated at a circular table. The number of ways, this can be done if A must have either B or C on his right and B must have either C or D on his right is
(a) 36
(b) 12
(c) 24
(d) 18
Answer: (d) 18

OTHER MODELS

Question. If \( n = 1! + 4! + 7! + .......... + 400! \) then ten's digit of 'n' is
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (b) 6

Question. Let \( a_n = \frac{10^n}{n!} \) for n = 1, 2, 3, ... . Then the greatest value of n for which \( a_n \) is the greatest is 
(a) 11
(b) 20
(c) 10
(d) 8
Answer: (c) 10

Question. If \( P(n, a) = P(n, b) \) where \( a < b \leq n \) (all are positive integers) and \( a + b = F n + G \) then \( |F| + |G| = \)
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (b) 3

Question. In a college of 300 students, every student read 5 newspapers and every newspaper is read by 60 students. The number of newspapers is
(a) at least 30
(b) at most 20
(c) exactly 25
(d) 24
Answer: (c) exactly 25

Question. The number of ways in which a TRUE or FALSE examination of 'n' statements can be answered on the asumption that no two consecutive questions are answered the same way is
(a) \( 2^{n-1} \)
(b) \( 2^n \)
(c) 1
(d) 2
Answer: (d) 2

Question. In a steamer there are 12 stalls for animals and there are horses, cows and calves (not less than 12 each) ready to be shipped. In how many ways can the ship load be made such that any stall can't have morethan one animal
(a) \( 3^{12} - 1 \)
(b) \( 3^{12} \)
(c) \( (12)^3 - 1 \)
(d) \( (12)^3 \)
Answer: (b) \( 3^{12} \)

Question. The number of different ways that three distinct rings can be worn to 4 fingers with atmost one ring in each of the fingers is
(a) \( ^4P_3 \)
(b) \( ^5P_4 \)
(c) \( ^5P_3 \)
(d) \( ^4P_2 \)
Answer: (a) \( ^4P_3 \)

Question. On a new year day every student of a class sends a card to every other student. The postman delivers 600 cards. The number of students in the class are
(a) 42
(b) 34
(c) 25
(d) 52
Answer: (c) 25

‘\( n^r \)’ MODEL

Question. The number of numbers greater than 1000 but not greater than 4000 that can be formed with the digits 0, 1, 2, 3, 4 repetition of digits being allowed is
(a) 375
(b) 376
(c) 377
(d) 378
Answer: (a) 375

Question. An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only three digits 2, 5 and 7. The smallest value of n for which this is possible is
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (b) 7

Question. The number of numbers less than 2000 that can be formed using the digits 1, 2, 3, 4 when repetition is allowed is
(a) 84
(b) 148
(c) 180
(d) 1440
Answer: (b) 148