TITLE: Binomial coefficient proof2
QUESTION [0 upvotes]: Having difficulty with starting off this proof.  Let n be a positive whole number. Prove that $$n\dbinom{2n}{n}=(n+1)\dbinom{2n}{n-1}$$. Any help would be greatly appreciated.

REPLY [3 votes]: Hint: $$
\binom{n}{k} = \frac{n!}{k!(n-k)!}
$$
details:
$$
n\binom{2n}{n}=n\frac{(2n)!}{n!n!}=
\frac{(2n)!}{n!(n-1)!} = (n+1)\frac{(2n)!}{(n+1)!(n-1)!}
= (n+1)\binom{2n}{n-1}
$$because $(n+1)+(n-1)=2n$.
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