combinatorics

Question

In how many ways can a group of n people be divided into pairs(2 people) ,given that n is an even number ?

Answer 1

    nC₂ * (n-2)C₂ * (n-4)C₂ * ... * (n-n+2)C₂ 

= $\frac{n!}{2!(n-2)!}$ * $\frac{(n-2)!}{2!(n-4)!}$ * ............ * $\frac{(n-n+2)!}{2!(n-n)!}$

= $\frac{n!}{(2!)^{\frac{n}{2}}}$

There will be $\frac{n}{2}$ pairs but since ordering of these pairs does not matter, the final answer comes out to be $\frac{n!}{2^{\frac{n}{2}}(\frac{n}{2})!}$

Answer 2

ANSWER IS (n-1)*(n-3)*(n-5)*.......1