(C) Given the equation: $^n{P_4} = 30 \times {^n}{C_5}$Using the formulas $^n{P_r} = \frac{n!}{(n-r)!}$ and $^n{C_r} = \frac{n!}{r!(n-r)!}$,we have:$\

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(C) Given the equation: $^n{P_4} = 30 \times {^n}{C_5}$Using the formulas $^n{P_r} = \frac{n!}{(n-r)!}$ and $^n{C_r} = \frac{n!}{r!(n-r)!}$,we have:$\frac{n!}{(n-4)!} = 30 \times \frac{n!}{5!(n-5)!}$Divide both sides by $n!$:$\frac{1}{(n-4)!} = \frac{30}{120 \times (n-5)!}$$\frac{1}{(n-4)(n-5)!} = \frac{1}{4 \times (n-5)!}$Equating the denominators:$n - 4 = 4$$n = 8$

Class 11 Mathematics Permutation and Combination Mix Examples-Permutation and Combination

Difficult

If $^n{P_4} = 30 \times {^n}{C_5}$,then $n = $

A
$6$
B
$7$
C
$8$
D
$9$

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Permutation and Combination

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Mix Examples-Permutation and Combination

The number of ordered pairs $(m, n)$,where $m, n \in \{1, 2, 3, \ldots, 50\}$,such that $6^m + 9^n$ is a multiple of $5$ is

KVPY 2012

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The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

Difficult

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DifficultKVPY 2019

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The digit in the unit place of the number $(183!) + (3^{183})$ is

Medium

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If $n$ is an integer with $0 \leq n \leq 11$,then the minimum value of $n!(11-n)!$ is attained when a value of $n$ equals to

MediumAP EAMCET 2014

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If $^n{P_4} = 30 \times {^n}{C_5}$,then $n = $

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