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

Q 11 :
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 12 :
The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

0%

(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 13 :
The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

0%

(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 14 :
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]

0%

(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 15 :
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 16 :
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 17 :
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 18 :
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 19 :
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 20 :
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.

Topic Question Set

Q 11 :
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 12 :
The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

0%

Q 13 :
The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

0%

Q 14 :
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]

0%

Q 15 :
The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

Q 16 :
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 17 :
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 18 :
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 19 :
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 20 :
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]

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Topic Question Set

Q 11 : 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 12 : The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

0%

Q 13 : The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

0%

Q 14 : 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]

0%

Q 15 : The largest n∈N such that 3n divides 50! is : [2025]

Q 16 : 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 17 : 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 18 : 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 19 : 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 20 : 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]

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Q 11 :
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 12 :
The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

Q 13 :
The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

Q 14 :
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 15 :
The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

Q 16 :
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 17 :
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 18 :
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 19 :
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 20 :
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]