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Question

# If $n$ is the number of ways of dividing $2n$ people into $n$ couples then

A

${2}^{n}N=\left(2n\right)!$

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B

$N\left(n!\right)=\left(1.3.5\dots \left(2n-1\right)\right)$

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C

$N={}^{2n}{C}_{n}$

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D

None of these

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Solution

## The correct option is A ${2}^{n}N=\left(2n\right)!$Explanation for correct answerNumber of ways of selecting r people from n people is given by, = ${}^{n}{C}_{r}$Number of ways, $N=\left({}^{2n}{C}_{n}\right)\left({}^{2n-2}C_{n}\right)\dots \dots .\left({}^{2}{C}_{2}\right)=\frac{\left(2n\right)!}{{2}^{n}}$ [we know, $\left({}^{2n}{C}_{n}\right)\left({}^{2n-2}C_{n}\right)\dots \dots .\left({}^{2}{C}_{2}\right)=\frac{\left(2n\right)!}{{2}^{n}}$]$\therefore {2}^{n}N=\left(2n\right)!\left[\because N=\frac{\left(2n\right)!}{{2}^{n}}\right]$Hence, option A is correct.

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