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

Q 61 :

The largest value of $n$ , for which $40^{n}$ divides $60!,$ is [2026]

13
14
12
11

1

2

3

4

(2)

$40^{n} = 2^{3 n} \times 5^{n}$

$E_{2} (60!) = [\frac{60}{2}] + [\frac{60}{2^{2}}] + [\frac{60}{2^{3}}] + [\frac{60}{2^{4}}] + [\frac{60}{2^{5}}]$

$= 30 + 15 + 7 + 3 + 1 = 56$

$E_{5} (60!) = [\frac{60}{5}] + [\frac{60}{5^{2}}]$

$= 12 + 2 = 14$

$40^{n} = {(2^{3})}^{n} \times 5^{n} = {(2^{3} \times 5)}^{n}$

$60! = 2^{56} \times 5^{14} . . . = 2^{14} \cdot {(2^{3} \cdot 5)}^{14}$

$∴$ $Maximum value of n is 14 .$

Q 62 :

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is [2026]

1579
1581
1580
1578

1

2

3

4

(3)

$ADIPRUU$

$A \to \frac{6!}{2!} = 360$

$D \to \frac{6!}{2!} = 360$

$P \to \frac{6!}{2!} = 360$

$R \to \frac{6!}{2!} = 360$

$U A \to 5! = 120$

$U D A P \to 3! = 6$

$U D A R \to 3! = 6$

$U D A U \to 3! = 6$

$U D A Y P R U \to 1$

$U D A Y P U R \to 1$

$Total = 1580$

Q 63 :

The largest $n \in N,$ for which $7^{n}$ divides $101!$ , is : [2026]

18
19
15
16

1

2

3

4

(4)

$Exponent of 7 in 101!$

$= [\frac{101}{7}] + [\frac{101}{7^{2}}] + [\frac{101}{7^{3}}] + \dots$

$= 14 + 2 = 16$

Q 64 :

Let S = {1,2,3,4,5,6,7,8,9}. Let $x$ be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let y be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then, [2026]

$45 x = 7 y$
$21 x = 4 y$
$29 x = 5 y$
$56 x = 9 y$

1

2

3

4

(2)

$S = {1, 2, 3, \dots, 9}$

$x = {}^{9}C_{1} \cdot {}^{8}C_{7} \times \frac{9!}{2} = \frac{9 \times 8 \times 9!}{2}$

$y = {}^{9}C_{2} \cdot {}^{7}C_{5} \times \frac{9!}{2! 2!} = \frac{9 \times 8}{2} \times \frac{7 \times 6}{2} \times \frac{9!}{2! 2!}$

$\Rightarrow \frac{x}{y} = \frac{4}{21}$

$\Rightarrow 21 x = 4 y$

Topic Question Set

Q 61 :

The largest value of $n$ , for which $40^{n}$ divides $60!,$ is [2026]

Q 62 :

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is [2026]

Q 63 :

The largest $n \in N,$ for which $7^{n}$ divides $101!$ , is : [2026]

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Topic Question Set

Q 61 : The largest value of n, for which 40n divides 60!, is [2026]

Q 62 : The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is [2026]

Q 63 : The largest n∈N, for which 7n divides 101!, is : [2026]

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Q 61 :

The largest value of $n$ , for which $40^{n}$ divides $60!,$ is [2026]

Q 62 :

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is [2026]

Q 63 :

The largest $n \in N,$ for which $7^{n}$ divides $101!$ , is : [2026]