(Análise Combinatória: Combinação – Coeficiente Binomial)
Veififique a identidade
[tex]\dfrac{1}{n-k}\dbinom{n-1}{k}=\dfrac{1}{n}\dbinom{n}{k}[/tex]
dados n, k naturais, com [tex]0\le k\le n-1.[/tex]
Veififique a identidade
[tex]\dfrac{1}{n-k}\dbinom{n-1}{k}=\dfrac{1}{n}\dbinom{n}{k}[/tex]
dados n, k naturais, com [tex]0\le k\le n-1.[/tex]
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Verified answer
[tex]\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}[/tex]
[tex]\boxed{\cfrac{1}{n-k}\dbinom{n-1}{k} = \cfrac{1}{n}\dbinom{n}{k}}[/tex]