Question

(Análise Combinatória: Combinação – Coeficiente Binomial)

Verifique a identidade

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

dados n, k naturais e $1\le k\le n.$​

Answer 1

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

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