Prove, by induction or otherwise, that

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

Find the number of sequences consisting of zeroes and ones that contain exactly $n$ zeroes and at most $m$ ones.