定理
\[\sum^n_{i=0}\dbinom{m+i}{i}=\dbinom{m+n+1}{m+1}
\]
证明
\[\begin{aligned}
\sum^n_{i=0}\dbinom{m+i}{i}&=\sum^n_{i=0}\dbinom{m+i}{m}\\
&=\dbinom{m}{m}+\dbinom{m+1}{m}+\dbinom{m+2}{m}+\dots+\dbinom{m+n}{m}\\
&=\dbinom{m+1}{m+1}+\dbinom{m+1}{m}+\dbinom{m+2}{m}+\dots+\dbinom{m+n}{m}\\
&=\dbinom{m+2}{m+1}+\dbinom{m+2}{m}+\dots+\dbinom{m+n}{m}\\
&\dots\\
&=\dbinom{m+n+1}{m+1}
\end{aligned}
\]