Question

Докажите тождество:
$C^{7}_{n} +C^{6}_{n} =C^{7}_{n+1}$

Answer 1

$C_{n}^{7}+C_{n}^{6} = \frac{n!}{7!(n-7)!} + \frac{n!}{6!(n-6)!} \\ (n-7)! = \frac{(n-6)!}{n-6} \\\\ \frac{n!}{7!(n-7)!} + \frac{n!}{6!(n-6)!} = \frac{n!(n-6)}{7!(n-6)!} + \frac{7n!}{7!(n-6)!} =$ 
 $\frac{n!n+n!}{7!(n-6)!} = \frac{n!(n+1)}{7(n-6)!} = \frac{ (n+1)!}{7! (n-6)!} = C_{n+1}^{7}$