 Hockey-stick identity

For nonnegative integers $n,m$, the binomial coefficients satisfyThe identity follows by mathematical induction: adding the next term combines two adjacent binomial coefficients through Pascal's identity. An equivalent form is ${\sum }_{r=j}^N\left(\genfrac{}{}{0ex}{}{r}{j}\right)=\left(\genfrac{}{}{0ex}{}{N+1}{j+1}\right)$, a diagonal sum in Pascal's triangle. It provides closed forms for sums of binomial coefficients without expanding individual factorials.

The "Hockey-stick identity" is a mathematical identity in combinatorics that describes a certain relationship involving binomial coefficients. It gets its name from the hockey stick shape that graphs of the identity can resemble.