# Help me prove that $\frac{(2n)!}{n!^2}$ is a natural number by induction. [duplicate]

When $n$ is a natural number, prove that $\frac{(2n)!}{n!^2}$ is a natural number.

If possible, I would like you to prove this by induction.

I tried to prove this by induction, but I can’t because $k+1$ is left in the denominator when I substitute $k+1$ to $n$.

Help me to solve this.

• please prove by induction – Gymnast Dec 6 '17 at 12:09
• – Barry Cipra Dec 6 '17 at 14:16

Doesn't this suffice? $$\frac{(2n)!}{n!^2} =\frac{(2n)!}{n!\,n!} =\binom{2n}{n}$$
• @GuyFsone, not if you use that $\binom{m}{n}$ is the number of subsets of size $n$ out of $m$ objects. – lhf Dec 6 '17 at 12:17