Question

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

• A. $2^{n}N=\left(2n\right)!$
• B. $N(n!)=(1.3.5\dots (2n-1\left)\right)$
• C. $N={}^{2n}{C}_n$
• D. None of these

Answer 1

The correct option is A

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

Explanation for correct answer

Number of ways of selecting r people from n people is given by, = ${}^n{C}_r$

Number of ways, $N=({}^{2n}{C}_n)({}^{2n-2}C_n)\dots \dots .({}^2{C}_2)=\frac{\left(2n\right)!}{2^n}$ [we know, $({}^{2n}{C}_n)({}^{2n-2}C_n)\dots \dots .({}^2{C}_2)=\frac{\left(2n\right)!}{2^n}$]

$\therefore 2^{n}N=\left(2n\right)!\left[\because N=\frac{\left(2n\right)!}{2^n}\right]$

Hence, option A is correct.