= Hockey-stick identity
{title2=$\sum_{r=j}^N\binom rj=\binom{N+1}{j+1}$}
For nonnegative <integers> $n,m$, the <binomial coefficients> satisfy
$$
\sum_{k=0}^m\binom{n+k}k=\binom{n+m+1}m.
$$
The identity follows by <mathematical induction>: adding the next term combines two adjacent <binomial coefficients> through <Pascal's identity>. An equivalent form is $\sum_{r=j}^N\binom rj=\binom{N+1}{j+1}$, a diagonal sum in <Pascal's triangle>. It provides closed forms for sums of <binomial coefficients> without expanding individual <factorials>.
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