Proving that a binomial coefficient is the sum of two others

Question

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?

http://math.stackexchange.com/questions/70190/simplifying-this-factorial-expression — pharmine, Nov 07 '11 at 17:43
Multiply top and bottom of first guy by $m$, of second guy by $n-m+1$. Add. — André Nicolas, Nov 07 '11 at 17:57

score 0 · Accepted Answer · answered Nov 07 '11 at 17:58

0

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

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

answered Nov 07 '11 at 17:58

Pedja

12,883

Leave something to the imagination; it is homework :-) – robjohn Nov 07 '11 at 18:40

Proving that a binomial coefficient is the sum of two others

1 Answers1