Numbers and Sets | Part IA, 2001

Prove, by induction or otherwise, that

(n0)+(n+11)++(n+mm)=(n+m+1m)\left(\begin{array}{l} n \\ 0 \end{array}\right)+\left(\begin{array}{c} n+1 \\ 1 \end{array}\right)+\cdots+\left(\begin{array}{c} n+m \\ m \end{array}\right)=\left(\begin{array}{c} n+m+1 \\ m \end{array}\right)

Find the number of sequences consisting of zeroes and ones that contain exactly nn zeroes and at most mm ones.

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