Problem - 3179
Show that $$\sum_{k=1}^n \binom{n}{k}\binom{n}{k-1}=\binom{2n}{n-1}$$
By the Vandermonde's identity, $$\sum_{k=1}^n\binom{n}{k}\binom{n}{k-1}=\sum_{i=0}^{n-1}\binom{n}{k+1}\binom{n}{k}=\sum_{k=0}^{n-1}\binom{n}{n-1-k}\binom{n}{k}=\binom{2n}{n-1}$$