朱世杰恒等式

${\displaystyle {\binom {m}{a+1}}+{\binom {m}{a}}+{\binom {m+1}{a}}...+{\binom {n}{a}}={\binom {n+1}{a+1}}}$

从${\displaystyle n}$元集${\displaystyle S=\{a_{1},a_{2},a_{3},...,a_{n}\}}$选${\displaystyle r}$个元素，有${\displaystyle {\binom {n}{r}}}$种方法。

必有${\displaystyle a_{1}}$时，在${\displaystyle n-1}$个元素中选${\displaystyle r-1}$个元素，排除${\displaystyle a_{1}}$，必有${\displaystyle a_{2}}$时，在${\displaystyle n-2}$个元素中选${\displaystyle r-1}$个元素，排除${\displaystyle a_{2}}$，如此类推，直到必有${\displaystyle a_{n-r+1}}$时，在${\displaystyle r-1}$个元素中选${\displaystyle r-1}$个元素。

${\displaystyle \sum _{k=r}^{n}{\binom {k-1}{r-1}}={\binom {n}{r}}}$[1]