Giải thích các bước giải:
Ta có:
$C^{k-1}_n+C^k_n$
$=\dfrac{n!}{(k-1)!\cdot (n-(k-1))!}+\dfrac{n!}{k!\cdot (n-k)!}$
$=\dfrac{n!}{(k-1)!\cdot (n-k+1)!}+\dfrac{n!}{k!\cdot (n-k)!}$
$=\dfrac{n!}{k!\cdot \dfrac1{k}\cdot (n-k)!\cdot (n-k+1)}+\dfrac{n!}{k!\cdot (n-k)!}$
$=\dfrac{n!\cdot \dfrac{k}{n-k+1}}{k!\cdot (n-k)!}+\dfrac{n!}{k!\cdot (n-k)!}$
$=\dfrac{n!}{k!\cdot (n-k)!}\cdot (\dfrac{k}{n-k+1}+1)$
$=\dfrac{n!}{k!\cdot (n-k)!}\cdot \dfrac{k+n-k+1}{n-k+1}$
$=\dfrac{n!}{k!\cdot (n-k)!}\cdot \dfrac{n+1}{n-k+1}$
$=\dfrac{n!\cdot (n+1)}{k!\cdot (n-k)!\cdot (n-k+1)}$
$=\dfrac{(n+1)!}{k!\cdot (n-k+1)!}$
$=\dfrac{(n+1)!}{k!\cdot ((n+1)-k)!}$
$=C^{k}_{n+1}$
