Question

(Análise Combinatória: Combinação – Coeficiente Binomial)

Veififique a identidade

     $\dfrac{1}{n-k}\dbinom{n-1}{k}=\dfrac{1}{n}\dbinom{n}{k}$

dados n, k naturais, com $0\le k\le n-1.$​

Answer 1

$\cfrac{1}{n-k}\dbinom{n-1}{k} \\\\= \cfrac{1}{n-k} \cdot \cfrac{(n-1)!}{k!(n-1-k)!}\\\\= \cfrac{(n-1)!}{k!(n-k)!}\\\\= \cfrac{n(n-1)!}{n \cdot k!(n-k)!}\\\\= \cfrac{1}{n} \cdot \cfrac{n!}{k!(n-k)!} \\\\= \cfrac{1}{n}\dbinom{n}{k}$

$\boxed{\cfrac{1}{n-k}\dbinom{n-1}{k} = \cfrac{1}{n}\dbinom{n}{k}}$