\input amstex \sum_{k \ge 0} \binom{n-k}{m-k} = \sum_{m-k \ge 0} \binom{n-(m-k)}{m-(m-k)}=\sum_{k \le m} \binom {n-m+k}{k}=\binom{n-m}{0}+\binom{n-m+1}{1}+\cdots + \binom{n-m+m}{m}=\binom{n-m+(m+1)}{m}=\binom{n+1}{m}