C k-1 n + C k n = C k n+1. Chứng minh đẳng thức trên

Giải thích các bước giải:

Ta có:

$C^{k-1}_n+C^k_n$

$=\dfrac{n!}{(k-1)!\cdot (n-(k-1))!}+\dfrac{n!}{k!\cdot (n-k)!}$

$=\dfrac{n!}{(k-1)!\cdot (n-k+1)!}+\dfrac{n!}{k!\cdot (n-k)!}$

$=\dfrac{n!}{k!\cdot \dfrac1{k}\cdot (n-k)!\cdot (n-k+1)}+\dfrac{n!}{k!\cdot (n-k)!}$

$=\dfrac{n!\cdot \dfrac{k}{n-k+1}}{k!\cdot (n-k)!}+\dfrac{n!}{k!\cdot (n-k)!}$

$=\dfrac{n!}{k!\cdot (n-k)!}\cdot (\dfrac{k}{n-k+1}+1)$

$=\dfrac{n!}{k!\cdot (n-k)!}\cdot \dfrac{k+n-k+1}{n-k+1}$

$=\dfrac{n!}{k!\cdot (n-k)!}\cdot \dfrac{n+1}{n-k+1}$

$=\dfrac{n!\cdot (n+1)}{k!\cdot (n-k)!\cdot (n-k+1)}$

$=\dfrac{(n+1)!}{k!\cdot (n-k+1)!}$

$=\dfrac{(n+1)!}{k!\cdot ((n+1)-k)!}$

$=C^{k}_{n+1}$