# Proving that a binomial coefficient is the sum of two others [duplicate]

I am asked to prove this given $1 \le m \le n - 1$ (homework question):

$$\frac{n!}{(m-1)!(n-m+1)!} + \frac{n!}{m!(n-m)!} = \frac{(n+1)!}{m!(n+1-m)!}$$

Which proof technique should I use to solve this? I tried induction but couldn't seem to get anywhere with it. Maybe I am just bad with factorial, should I use induction?

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## marked as duplicate by sandwich, Daniel W. Farlow, Servaes, Michael Medvinsky, Najib IdrissiDec 6 '15 at 12:16

This question was marked as an exact duplicate of an existing question.

Multiply top and bottom of first guy by $m$, of second guy by $n-m+1$. Add. – André Nicolas Nov 7 '11 at 17:57

$$\frac{n!\cdot m}{m(m-1)!(n+1-m)!}+\frac{n!\cdot (n+1-m)}{m!(n+1-m)!}=\frac{m \cdot n!+(n+1)n!-m\cdot n!}{m!(n+1-m)!}=$$
$$=\frac{(n+1)!}{m!(n+1-m)!}$$