Permutations and Combinations jee mains questions | JEE Main Maths Permutations and Combinations Previous Year Question

Q 51 :

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is __________. [2025]

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(17280)

Number of ways that all boys sit together $= 5! \times 5!$

Number of ways no two boys sit together $= 4! \times 5!$

$∴$ Required number of ways $= 5! \times 5! + 4! \times 5! = 17280$

Q 52 :

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is __________. [2025]

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(125)

Number of 3-digits = 999 – 99 = 900

Number of 3-digit numbers divisible by 2 & 3 i.e., by 6,

$\frac{900}{6} = 150$

Number of 3-digit numbers divisible by 4 & 9 i.e., by 36,

$\frac{900}{36} = 25$

$∴$ Number of 3-digit numbers divisible by 2 & 3 but not 4 & 9 = 150 – 25 = 125.

Q 53 :

Number of functions $f : {1, 2, . . ., 100} \to {0, 1}$ , that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________. [2025]

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(392)

Given : $f : {1, 2, . . ., 100} \to {0, 1}$

Number of ways to connect {1, 2, ..., 98} to 1 = 98

Number 99 can connect either 0 or 1 $\Rightarrow$ 2 ways

Similarly, 100 can connect either 0 or 1 $\Rightarrow$ 2 ways

$∴$ Total number of functions for the given condition that assign 1 to exactly one of positive integers $\leq$ 98 is given by $98 \times 2 \times 2$ = 392.

Q 54 :

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is __________. [2025]

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(64)

Let xyz be any number between 212 and 999

Let $x = 2 \to y + z = 13$ , then

(y, z) : (4, 9), (5, 8), (6, 7), (7, 6), (8, 5), (9, 4), i.e., 6 in number.

Let $x = 3 \to y + z = 12$ , then

(y, z) : (3, 9), (4, 8), (5, 7), (6, 6), (7, 5), (8, 4), (9, 3) i.e., 7 in number.

Let $x = 4 \to y + z = 11$ , then

(y, z) : (2, 9), (3, 8), (4, 7), (5, 6), (6, 5), (7, 4),(8, 3), (9, 2) i.e., 8 in number.

Let $x = 5 \to y + z = 10$ , then

(y, z) : (1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3). (8, 2), (9, 1) i.e., 9 in number.

Let $x = 6 \to y + z = 9$ , then

(y, z) : (0, 9), (1, 8), (2, 7), (3, 6), (4, 5), (5, 4), (6, 3), (7, 2), (8, 1), (9, 0) i.e., 10 in number.

Let $x = 7 \to y + z = 8$ , then

(y, z) : (0, 8), (1, 7), (2, 6), (3, 5), (4, 4), (5, 3), (6, 2), (7, 1), (8, 0) i.e., 9 in number.

Let $x = 8 \to y + z = 7$ , then

(y, z) : (0, 7), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6,1), (7, 0) i.e., 8 in number.

Let $x = 9 \to y + z = 6$ , then

(y, z) : (0, 6), (1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0) i.e., 7 in number.

Total = 6 + 7 + 8 + 9 + 10 + 9 + 8 + 7 = 64.

Q 55 :

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is __________. [2026]

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(1422)

$P \to 3, Q \to 2, R \to 2, S, T, U, V$

$Case I: 3 alike, 1 different$

${}^{1}C_{1} \times {}^{6}C_{1} \times \frac{4!}{3!} = 24$

$Case II: 2 alike, 2 alike$

${}^{3}C_{2} \times \frac{4!}{2! 2!} = 18$

$Case III: 2 alike, 2 different$

${}^{3}C_{1} \times {}^{6}C_{2} \times \frac{4!}{2!} = 540$

$Case IV: All 4 different$

${}^{7}C_{4} \times 4! = 840$

$Total words = 1422$

Q 56 :

The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to: [2026]

64
256
16
1024

1

2

3

4

(1)

Number of relation which are reply and sym. both $= 1^{4} \times 2^{6} = 64$

(a, a) (a, b) (a, c) (a, d)
(b, a) (b, b) (b, c) (b, d)
(c, a) (c, b) (c, c) (c, d)
(d, a) (d, b) (d, c) (d, d)

Q 57 :

The number of strictly increasing functions f from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq i$ for $1 \leq i \leq 6,$ is equal to [2026]

28
22
27
21

1

2

3

4

(1)

$f (i) \neq i, f (x) is strictly increasing function$

$f : A \to B,$ $where A = {1, 2, 3, \dots, 6}$

$B = {1, 2, 3, \dots, 9},$ then number of functions $f : A \to B$ is equal to

$f (i) \neq i$ $Case (i) f (1) = 2 \Rightarrow {}^{7}{C_{5}} = 21$

$Case (ii) f (1) = 3 \Rightarrow {}^{6}{C_{5}} = 6$

$Case (iii) f (1) = 4 \Rightarrow {}^{5}{C_{5}} = 1$

Number of functions from A to B $= 21 + 6 + 1 = 28$

Q 58 :

The number of $3 \times 2$ matrices A, which can be formed using the elements of the set ${- 2, - 1, 0, 1, 2}$ such that the sum of all the diagonal elements of $A^{T} A$ is 5, is ________ . [2026]

0%

(312)

${(\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix})}_{3 \times 2}$

$A^{T} A = {(\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix})}_{2 \times 3} {(\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix})}_{3 \times 2}$

$= (\begin{matrix} a_{1}^{2} + a_{2}^{2} + a_{3}^{2} & \dots \\ \dots & b_{1}^{2} + b_{2}^{2} + b_{3}^{2} \end{matrix})$

$Tr (A^{T} A) = a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + b_{1}^{2} + b_{2}^{2} + b_{3}^{2} = 5$

${2, 1, 0, 0, 0, 0}$

${2, - 1, 0, 0, 0, 0}$

${- 2, 1, 0, 0, 0, 0}$

${- 2, - 1, 0, 0, 0, 0}$

${1, 1, 1, 1, 1, 0}$

$No. of ways = \frac{6!}{4!} \times 4 + 2 \times \frac{6!}{5!} + 2 \times \frac{6!}{4!} + 2 \times \frac{6!}{3! 2!}$

$= \frac{6!}{3!} + 2 \times 6 \times 2 + 2 \times 15 \times 2 \times \frac{6!}{3!}$

$= 120 + 120 + 12 + 60 = 312$

Q 59 :

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______. [2026]

0%

(42)

$(1) all different$

$5, 0, 1, 9 \Rightarrow {}^{4}C_{3} = 6 ways$

$(2) 2 alike, 2 different$

$5, 0, 0, 1 \Rightarrow 3 ways$

$5, 1, 1, 2 \Rightarrow 3 ways$

$5, 2, 2, 0 \Rightarrow 3 ways$

$5, 2, 2, 9 \Rightarrow 3 ways$

$5, 5, 0, 2 \Rightarrow 6 ways$

$5, 5, 2, 9 \Rightarrow 6 ways$

$5, 1, 9, 9 \Rightarrow 3 ways$

$(3) 3 alike, 1 different$

$5, 5, 5, 0 \Rightarrow 3 ways$

$5, 5, 5, 9 \Rightarrow 3 ways$

$(4) 2 alike, 2 other alike$

$5, 5, 1, 1 \Rightarrow 3 ways$

$Total ways = 42$

Q 60 :

Three persons enter in a lift at the ground floor. The lift will go upto 10th floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ________. [2026]

0%

(210)

${}^{7}C_{3} \times 3! = \frac{7 \times 6 \times 5}{1 \times 2 \times 3} \times 3! = 210$

Topic Question Set

Q 51 :

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is __________. [2025]

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Q 52 :

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is __________. [2025]

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Q 53 :

Number of functions $f : {1, 2, . . ., 100} \to {0, 1}$ , that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________. [2025]

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Q 54 :

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is __________. [2025]

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Q 55 :

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is __________. [2026]

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Q 56 :

The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to: [2026]

Q 57 :

The number of strictly increasing functions f from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq i$ for $1 \leq i \leq 6,$ is equal to [2026]

Q 58 :

The number of $3 \times 2$ matrices A, which can be formed using the elements of the set ${- 2, - 1, 0, 1, 2}$ such that the sum of all the diagonal elements of $A^{T} A$ is 5, is ________ . [2026]

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Q 59 :

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______. [2026]

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Q 60 :

Three persons enter in a lift at the ground floor. The lift will go upto 10th floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ________. [2026]

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Topic Question Set

Q 51 : The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is __________. [2025]

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Q 52 : The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is __________. [2025]

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Q 53 : Number of functions f:{1,2,...,100}→{0,1}, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________. [2025]

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Q 54 : The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is __________. [2025]

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Q 55 : The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is __________. [2026]

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Q 56 : The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to: [2026]

Q 57 : The number of strictly increasing functions f from the set {1,2,3,4,5,6} to the set {1,2,3,…,9} such that f(i)≠i for 1≤i≤6, is equal to [2026]

Q 58 : The number of 3×2 matrices A, which can be formed using the elements of the set {-2,-1,0,1,2} such that the sum of all the diagonal elements of ATA is 5, is ________ . [2026]

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Q 59 : The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______. [2026]

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Q 60 : Three persons enter in a lift at the ground floor. The lift will go upto 10th floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ________. [2026]

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Q 51 :

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is __________. [2025]

Q 52 :

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is __________. [2025]

Q 53 :

Number of functions $f : {1, 2, . . ., 100} \to {0, 1}$ , that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________. [2025]

Q 54 :

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is __________. [2025]

Q 55 :

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is __________. [2026]

Q 56 :

The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to: [2026]

Q 57 :

The number of strictly increasing functions f from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq i$ for $1 \leq i \leq 6,$ is equal to [2026]

Q 58 :

The number of $3 \times 2$ matrices A, which can be formed using the elements of the set ${- 2, - 1, 0, 1, 2}$ such that the sum of all the diagonal elements of $A^{T} A$ is 5, is ________ . [2026]

Q 59 :

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______. [2026]

Q 60 :

Three persons enter in a lift at the ground floor. The lift will go upto 10th floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ________. [2026]