سلام. وقت بخیر.

$S_{m,n}=\sum_{k=0}^m\frac{\binom{m}{k}}{\binom{n}{k}}=\sum_{k=0}^m\frac{\frac{m!​}{(m-k)!k!}}{\frac{n!}{(n-k)!k!}}$

$=\sum_{k=0}^m\frac{\frac{(n-k)!}{(m-k)!k!}}{\frac{n!}{m!k!}}=\sum_{k=0}^m\frac{\frac{(n-k)!}{(m-k)!(n-m)!}}{\frac{n!}{m!(n-m)!}}=\sum_{k=0}^m\frac{\binom{n-k}{n-m}}{\binom{n}{m}}=\frac{\binom{n+1}{m}}{\binom{n}{m}}$