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

Q 11 :

The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is _______ . [2023]

(240)

Given digits, 1, 2, 2, 2, 3, 3, 5

Total digits = 7

Total number of seven digits $= \frac{7!}{3! 2!} = 420$

Total number of seven digit even numbers $= \frac{3 \times 6!}{2! 3!} = 180$

$∴$ Total number of seven digit odd numbers = 420 - 180 = 240

Q 12 :

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is _______ . [2023]

(2997)

Number starting with 2 and 3 = $2 \times 6^{4}$

Number starting with 40 = $6^{3}$

Number starting with 420, 422, 423, 424, 427 = $5 \times 6^{2}$

Similarly, number starting with 4290 = 6

$Total = 2 \times 6^{4} + 6^{3} + 5 \times 6^{2} + 6 + 3 = 2997$

Q 13 :

If ${}^{2 n + 1}{P_{n - 1} : {}^{2 n - 1}{P_{n} = 11 : 21,}}$ then $n^{2} + n + 15$ is equal to ______ . [2023]

(45)

Given, ${}^{2 n + 1}{P_{n - 1}} :^{2 n - 1} P_{n} = 11 : 21$

$\Rightarrow \frac{(2 n + 1)!}{(2 n + 1 - n + 1)!} \times \frac{(n - 1)!}{(2 n - 1)!} = \frac{11}{21}$

$\Rightarrow \frac{(2 n + 1)!}{(n + 2)!} \times \frac{(n - 1)!}{(2 n - 1)!} = \frac{11}{21}$

$\Rightarrow \frac{(2 n + 1) (2 n) (2 n - 1)!}{(n + 2) (n + 1) (n) (n - 1)!} \times \frac{(n - 1)!}{(2 n - 1)!} = \frac{11}{21}$

$\Rightarrow \frac{(2 n + 1) (2 n)}{(n + 2) (n + 1) (n)} = \frac{11}{21} \Rightarrow \frac{4 n + 2}{n^{2} + 3 n + 2} = \frac{11}{21}$

$\Rightarrow 84 n + 42 = 11 n^{2} + 33 n + 22 \Rightarrow 11 n^{2} - 51 n - 20 = 0$

$\Rightarrow (n - 5) (11 n + 4) = 0 \Rightarrow n = 5, \frac{- 4}{11}$

The value of $n$ can never be negative.

So, $n = 5$

$∴$ $n^{2} + n + 15 = {(5)}^{2} + 5 + 15 = 45$

Q 14 :

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is [2024]

NRAPGU
NRAGPU
NRAPUG
NRAGUP

1

2

3

4

(1)

Letter in the word NAGPUR = 6

If we fix A _ _ _ _ _

Number of words start with A = 5! = 120

If we fix G _ _ _ _ _

Number of words start with G = 5! = 120

If we fix N A _ _ _ _

Number of words start with NA = 4! = 24

If we fix N G _ _ _ _

Number of words start with NG = 4! = 24

If we fix N P _ _ _ _

Number of words start with NP = 4! = 24

Next word i.e., $313^{th}$ word in dictionary will be NRAGPU, $314^{th}$ word will be NRAGUP, $315^{th}$ word will be NRAPGU.

Q 15 :

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

OBBJH
HBBJO
OBBHJ
JBBOH

1

2

3

4

(1)

Let us fix B first (Dictionary Order)

B _ _ _ _ (We have 4 distinct letters left, to be arranged in 4 distinct ways)
Number of ways = 4! = 24

H _ _ _ _ (We have B, B, J, O to be arranged in 4 ways when H is fixed)

Number of ways = $\frac{4!}{2!} = 12$

J _ _ _ _ (B, B, H, O to be arranged in 4 ways)

Number of ways = $\frac{4!}{2!} = 12$

49th word will be OBBHJ
50th word will be OBBJH

Q 16 :

If $n$ is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to [2024]

43
47
53
51

1

2

3

4

(4)

Number of ways in which five different employees can sit are

$I$ $I I$ $I I I$ $I V$

$□$ $□$ $□$ $□$

5 0 0 0 $\to$ 5!/5! = 1

4 1 0 0 $\to$ 5!/4! = 5

3 2 0 0 $\to$ 5!/3!2! = 10

3 1 1 0 $\to$ 5!/3! 1! 1! 2! = 10

2 2 1 0 $\to$ $\frac{5!}{2! 2! 1! 2!} = 15$

2 1 1 1 $\to$ $\frac{5!}{2! 1! 1! 1! 3!} = 10$

$∴$ Total number of ways =1 + 5 + 10 + 10 + 15 + 10 = 51

Q 17 :

Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then

$α \notin N$ and $β \in N$
$α \in N$ and $β \notin N$
$α \in N$ and $β \in N$
$α \notin N$ and $β \notin N$

1

2

3

4

(3)

Q 18 :

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

18
15
12
16

1

2

3

4

(2)

Ways of arranging 8 identical books in 4 identical shelves = {8, 0, 0, 0}, {7, 1, 0, 0}, {6, 2, 0, 0}, {5, 3, 0, 0}, {4, 4, 0, 0}, {6, 1, 1, 0},{5, 2, 1, 0}, {4, 3, 1, 0}, {4, 2, 2, 0}, {3, 3, 2, 0}, {5, 1, 1, 1}, {4, 2, 1, 1}, {3, 3, 1, 1}, {3, 2, 2, 1}, {2, 2, 2, 2}.

Number of ways = 15

[As shelves and books are identical, so the arrangements (5, 2, 1, 0), (2, 5, 1, 0), and (1, 2, 5, 0) are considered as same]

Q 19 :

The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

(125)

We need sum of 16 on throwing a dice four times so possible ways are:
{(4, 4, 4, 4), (5, 4, 4, 3), (5, 5, 1, 5), (5, 5, 3, 3), (5, 5, 4, 2), (6, 4, 3, 3), (6, 4, 4, 2), (6, 5, 3, 2), (6, 5, 4, 1), (6, 6, 3, 1), (6, 6, 2, 2)}

Obtained Result	Number of Ways
(6, 6, 2, 2)	$\frac{4!}{2! 2!} = 6$
(6, 6, 3, 1)	$\frac{4!}{2!} = 12$
(6, 5, 4, 1)	$4! = 24$
(6, 5, 3, 2)	$4! = 24$
(6, 4, 4, 2)	$\frac{4!}{2!} = 12$
(6, 4, 3, 3)	$\frac{4!}{2!} = 12$
(5, 5, 4, 2)	$\frac{4!}{2!} = 12$
(5, 5, 3, 3)	$\frac{4!}{2! 2!} = 6$
(5, 5, 1, 5)	$\frac{4!}{3!} = 4$
(5, 4, 4, 3)	$\frac{4!}{2!} = 12$
(4, 4, 4, 4)	$1! = 1$
Total ways	125

Q 20 :

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

(70)

Number between 100 to 1000 are 3-digit number.

Let number $a b c$ such that $a + b + c = 14$

where $a, b, c \in {0, 1, 2, . . . 9}$ and $a \geq 1$

Case I : All three digits are same ie., $a = b = c$

$\Rightarrow 3 a = 14$ , which is not possible

Case II : Two digits are same i.e., $a = b \neq c$

$\Rightarrow 2 a + c = 14$

$∴$ (a, c) = {(3, 8), (4, 6), (5, 4), (6, 2), (7, 0)}

In each subcase, number of ways of forming a 3-digit number = $\frac{3!}{2!}$

There are 5 such cases as (a, c) have 5 elements in which 0,7,7 is included

So, total number of 3 digits numbers when 2 digits are same $5 \times \frac{3!}{2!} - 1$

$= 15 - 1 = 14$

Case III : All digits are different

$∴$ (a, b, c) = {(1, 4, 9), (2, 4, 8), (2, 3, 9), (1, 5, 8), (3, 4, 7), (2, 5, 7), (1, 6, 7), (3, 5, 6), (5, 9, 0), (6, 8, 0)}

Total number of 3 digits number formed by above triplet = $10 \times 3! - 2 \times 2! = 60 - 4 = 56$

Total number of required number = 56 + 14 = 70

Q 21 :

All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]

(553)

Number of words starts with $E = \frac{6!}{2!}$

Number of words starts with $G E = \frac{5!}{2!}$

Number of words starts with $G N = \frac{5!}{2!}$

Number of words starts with $GTE = 4!$

Number of words starts with $GTN = 4!$

Number of words starts with $GTT = 4!$

Next word will be GTWENTY

$∴$ Rank of GTWENTY $= \frac{6!}{2!} + 2 \times \frac{5!}{2!} + 3 \times 4! + 1 = 553$

Q 22 :

The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

23
22
21
20

1

2

3

4

(2)

The number of times a prime p divides $n!$ is $\sum_{k = 1}^{\infty} [\frac{n}{p^{k}}]$

Required largest value of

$n = [\frac{50}{3}] + [\frac{50}{3^{2}}] + [\frac{50}{3^{3}}]$

= 16 + 5 + 1 = 22

Hence, the maximum value of $n$ is 22.

Q 23 :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to : [2025]

360
1820
45
2520

1

2

3

4

(4)

Given number in the sequences are 11111 222 00

$∴$ Number of sequences = $\frac{10!}{5! 3! 2!}$ = 2520

Q 24 :

In a group of 3 girls and 4 boys, there are two boys $B_{1}$ and $B_{2}$ . The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_{1}$ and $B_{2}$ are not adjacent to each other, is : [2025]

144
120
96
72

1

2

3

4

(1)

Total number of ways

$= 3! \times 4! \times 2! - 3! \times 3! \times 2! \times 2! = 144$

Q 25 :

The Number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is [2025]

34000
36000
37000
35000

1

2

3

4

(2)

Number of words in which vowels never come together

= Total number of words – Number of words in which vowels come together

$= 8! - 6! \times 3!$ = 36000

Q 26 :

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is [2025]

4608
4607
5719
5720

1

2

3

4

(2)

Given digits are 0, 1, 2, 3, 4, 5, 6, 7.

First place can be filled in 3ways, i.e., (5, 6, 7)

For $1^{st}$ place as 5, last place can be 0, 1, 2, 3

$∴$ Total number of ways = $1 \times 512 \times 4$ = 2048

For $1^{st}$ place as 6, last place can be 0, 1, 2

$∴$ Total number of ways = $1 \times 512 \times 3$ = 1536

For $1^{st}$ place as 7, last place can be 0, 1

Total number of ways = $1 \times 512 \times 2$ = 1024

$∴$ Total number of ways = 2048 + 1536 + 1024 = 4608

So, 50000 is not included i.e., 4608 – 1 = 4607

Q 27 :

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 : [2025]

158
164
173
161

1

2

3

4

(4)

(i) Number of numbers formed using 1 and 3, i.e.,

$= \frac{7!}{5! 2!} = 21$

(ii) Number of numbers formed using 1, 2, 3 i.e.,

$= \frac{7!}{4! 2!} = 105$

(iii) Number of numbers formed using 1 and 2, i.e.,

$= \frac{7!}{3! 4!} = 35$

Number of elements in set P = 161.

Q 28 :

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $440^{th}$ position in this arrangement, is : [2025]

PRNAUK
PRKANU
PRKAUN
PRNAKU

1

2

3

4

(3)

We have, A, K, N, P, R, U

Now, to calculate rank

Number of words start with A = 5! = 120

Number of words starts with K = 5! = 120

Number of words starts with N = 5! = 120

Number of words starts with PA = 4! = 24

Number of words starts with PK = 4! = 24

Number of words starts with PN = 4! = 24

Number of words starts with PRA = 3! = 6

Number of words starts with PRKANU = 1

Number of words starts with PRKAUN = 1

$Total = 440$

$∴ 440^{th}$ words is PRKAUN.

Q 29 :

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^{n}; m, n \in {1, 2, 3, . . ., 9}$ , then m + n is equal to __________. [2025]

(14)

Total 7 digit numbers = 9000000

7 digit numbers having sum of their digits is even

$= \frac{9000000}{2} = 9 \cdot 5 \cdot 10^{5}$

On comparing with $m \cdot n \cdot 10^{n}$ , we get

m = 9 and n = 5

$∴$ m + n = 14.

Q 30 :

Let m and n, (m < n), be two 2-digit numbers. Then the total numbers of pairs (m, n), such that gcd (m, n) = 6, is __________. [2025]

(64)

Given m and n are 2-digit numbers (m < n).

Such that gcd (m, n) = 6.

Let m = 6a, n = 6b, where a and b are coprime integers a < b.

$\Rightarrow 10 \leq m \leq 99, 10 \leq n \leq 99$ .

$\Rightarrow 2 \leq a \leq 16, 2 \leq b \leq 16$ .

Now, if a = 2, then b = 3, 5, 7, 9, 11, 13, 15 = 7

a = 3, then b = 4, 5, 7, 8, 10, 11, 13, 14, 16 = 9

a = 4, then b = 5, 7, 9, 11, 13, 15 = 6

a = 5, then b = 6, 7, 8, 9, 11, 12, 13, 14, 16 = 9

a = 6, then b = 7, 11, 13 = 3

a = 7, then b = 8, 9, 10, 11, 12, 13, 15, 16 = 8

a = 8, then b = 9, 11, 13, 15 = 4

a = 9, then b = 10, 11, 13, 14, 16 = 5

a = 10, then b = 11, 13 = 2

a = 11, then b = 12, 13, 14, 15, 16 = 5

a = 12, then b = 13 = 1

a = 13, then b = 14, 15, 16 = 3

a = 14, then b = 15 = 1

a = 15, then b = 16 = 1

$∴$ Total possible number of ordered pairs = 64.

Q 31 :

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]

(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 32 :

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

(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 33 :

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]

(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 34 :

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

(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 35 :

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

(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 36 :

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 37 :

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 38 :

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]

(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 39 :

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]

(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 40 :

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]

(210)

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

Topic Question Set

Q 11 :

The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is _______ . [2023]

Q 12 :

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is _______ . [2023]

Q 13 :

If ${}^{2 n + 1}{P_{n - 1} : {}^{2 n - 1}{P_{n} = 11 : 21,}}$ then $n^{2} + n + 15$ is equal to ______ . [2023]

Q 14 :

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is [2024]

Q 15 :

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

Q 16 :

If $n$ is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to [2024]

Q 17 :

Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then

(3)

Q 18 :

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

Q 19 :

The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

Q 20 :

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

Q 21 :

All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]

Q 22 :

The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

Q 23 :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to : [2025]

Q 24 :

In a group of 3 girls and 4 boys, there are two boys $B_{1}$ and $B_{2}$ . The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_{1}$ and $B_{2}$ are not adjacent to each other, is : [2025]

Q 25 :

The Number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is [2025]

Q 26 :

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is [2025]

Q 27 :

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 : [2025]

Q 28 :

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $440^{th}$ position in this arrangement, is : [2025]

Q 29 :

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^{n}; m, n \in {1, 2, 3, . . ., 9}$ , then m + n is equal to __________. [2025]

Q 30 :

Let m and n, (m < n), be two 2-digit numbers. Then the total numbers of pairs (m, n), such that gcd (m, n) = 6, is __________. [2025]

Q 31 :

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 32 :

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

Q 33 :

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 34 :

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

Q 35 :

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

Q 36 :

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

Q 37 :

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 38 :

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 39 :

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 40 :

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 11 : The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is _______ . [2023]

Q 12 : Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is _______ . [2023]

Q 13 : If Pn-1:Pn=11:21,2n-12n+1 then n2+n+15 is equal to ______ . [2023]

Q 14 : If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at 315th position in this arrangement is [2024]

Q 15 : 60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

Q 16 : If n is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to [2024]

Q 17 : Let α=(4!)!(4!)3! and β=(5!)!(5!)4!. Then

(3)

Q 18 : Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

Q 19 : The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

Q 20 : The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

Q 21 : All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]

Q 22 : The largest n∈N such that 3n divides 50! is : [2025]

Q 23 : The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to : [2025]

Q 24 : In a group of 3 girls and 4 boys, there are two boys B1 and B2. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but B1 and B2 are not adjacent to each other, is : [2025]

Q 25 : The Number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is [2025]

Q 26 : The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is [2025]

Q 27 : 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 : [2025]

Q 28 : If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440th position in this arrangement, is : [2025]

Q 29 : If the number of seven-digit numbers, such that the sum of their digits is even, is m·n·10n; m, n∈{1,2,3,...,9}, then m + n is equal to __________. [2025]

Q 30 : Let m and n, (m < n), be two 2-digit numbers. Then the total numbers of pairs (m, n), such that gcd (m, n) = 6, is __________. [2025]

Q 31 : 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 32 : The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is __________. [2025]

Q 33 : 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]

Q 34 : The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is __________. [2025]

Q 35 : The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is __________. [2026]

Q 36 : The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to: [2026]

Q 37 : 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 38 : 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]

Q 39 : 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 40 : 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 11 :

The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is _______ . [2023]

Q 12 :

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is _______ . [2023]

Q 13 :

If ${}^{2 n + 1}{P_{n - 1} : {}^{2 n - 1}{P_{n} = 11 : 21,}}$ then $n^{2} + n + 15$ is equal to ______ . [2023]

Q 14 :

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is [2024]

Q 15 :

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

Q 16 :

If $n$ is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to [2024]

Q 17 :

Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then

Q 18 :

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

Q 19 :

The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

Q 20 :

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

Q 21 :

All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]

Q 22 :

The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

Q 23 :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to : [2025]

Q 24 :

In a group of 3 girls and 4 boys, there are two boys $B_{1}$ and $B_{2}$ . The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_{1}$ and $B_{2}$ are not adjacent to each other, is : [2025]

Q 25 :

The Number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is [2025]

Q 26 :

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is [2025]

Q 27 :

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 : [2025]

Q 28 :

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $440^{th}$ position in this arrangement, is : [2025]

Q 29 :

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^{n}; m, n \in {1, 2, 3, . . ., 9}$ , then m + n is equal to __________. [2025]

Q 30 :

Let m and n, (m < n), be two 2-digit numbers. Then the total numbers of pairs (m, n), such that gcd (m, n) = 6, is __________. [2025]

Q 31 :

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 32 :

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

Q 33 :

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 34 :

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

Q 35 :

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

Q 36 :

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

Q 37 :

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 38 :

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 39 :

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 40 :

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]