Question

Demonstraţi relaţia valabila pt orice n,k apartine N stelat, k<n.​

Answer 1

Răspuns:

$= \frac{k+1}{n-k}$

Explicație pas cu pas:

Formula pentru combinări este:

$C_{n} ^{k} = \frac{n!}{k!(n-k)!}$

O altă formulă care ne ajută este n! = (n-1)!·n

$\frac{C_{n} ^{k} }{C_{n} ^{k+1} } = \frac{\frac{n!}{k!(n-k)!} }{\frac{n!}{(k+1)!(n-k-1)!} } = \frac{n!}{k!(n-k)!} *\frac{(k+1)!(n-k-1)!}{n!}$

$= \frac{k!(k+1)(n-k-1)!}{k!(n-k-1)!*(n-k)} = \frac{k+1}{n-k}$

Answer 2

$\it \dfrac{C^k_n}{C^{k+1}_n}=C^k_n\cdot\dfrac{1}{C^{k+1}_n}=\dfrac{n!}{k!(n-k)!}\cdot\dfrac{(k+1)!(n-k-1)!}{n!}=\\ \\ \\ =\dfrac{k!(k+1)(n-k-1)!}{k!(n-k-1)(n-k)}=\dfrac{k+1}{n-k}$