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

Q 41 :
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 42 :
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 43 :
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 44 :
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 45 :
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 46 :
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 47 :
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 48 :
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 49 :
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]

0%

(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 50 :
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]

0%

(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.

Topic Question Set

Q 41 :
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 42 :
The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

Q 43 :
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 44 :
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 45 :
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 46 :
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 47 :
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 48 :
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 49 :
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]

0%

Q 50 :
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]

0%

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Topic Question Set

Q 41 : 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 42 : The largest n∈N such that 3n divides 50! is : [2025]

Q 43 : 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 44 : 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 45 : 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 46 : 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 47 : 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 48 : 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 49 : 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]

0%

Q 50 : 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]

0%

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Q 41 :
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 42 :
The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

Q 43 :
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 44 :
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 45 :
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 46 :
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 47 :
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 48 :
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 49 :
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 50 :
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]