I am trying to prove $\sum_{j=1}^{n-k}{n \choose j}{n-k-1 \choose j-1}={2n-k-1 \choose n-k}$. I tried to apply Vandermonde's Identity, however I have not been able to.
2026-04-09 00:25:06.1775694306
Sum of the product of binomial coefficients
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If you rewrite the left-hand side as $$\sum_{j=1}^{n-k}\binom{n}{j} \binom{n-k-1}{n-k-j},$$ you can then use Vandermonde's identity.