Is $\frac{n!}{\left(n/2\right)!\left(n/2\right)!}$ a natural number for $n$ even?

Question

Probably this is a easy question, but I was unable to solve it. Let $n$ be a even natural number. Is true that the following number is natural for all $n$? $$\frac{n!}{\left(n/2\right)!\left(n/2\right)!}$$

I can see, for example, that $(n/2)!$ divides $n!$, but then I can't conclude by using this arguing.

Thank you

Answer 1

Yes, it is. Suppose that $n=2m$; then

$$\frac{n!}{(n/2)!(n/2)!}=\frac{(2m)!}{m!m!}=\binom{2m}m\;,$$

which is the number of $m$-element subsets of a set of $2m$ things. This clearly must be an integer. For more information, see the Wikipedia article on binomial coefficients.

Answer 2

Yes. This is just $\displaystyle {n \choose n/2}$.