Counting binary strings with bounded ones ($\#\{w:\#0(w)=n,\#1(w)\le m\}=\left(\genfrac{}{}{0ex}{}{n+m+1}{m}\right)$)

A binary string with $n$ zeroes and $j$ ones is determined by its $j$ one-positions, giving $\left(\genfrac{}{}{0ex}{}{n+j}{j}\right)$ possibilities. Sum for $0\le j\le m$ and apply the hockey-stick identity. Equivalently, distribute at most $m$ ones among the $n+1$ gaps around the zeroes and add a slack count; stars and bars counts $n+2$ nonnegative counts summing to $m$.