Question

Sa se verifice egalitatea. Exercițiul 6 - subpunctul b)
Clasa a 10-a , combinari

Answer 1

$C_{n}^k = \dfrac{n!}{k!(n-k)!} = \dfrac{(n-1)!\cdot n}{k!\Big((n-1)-k\Big)!\cdot (n-k) }= \dfrac{n}{n-k}C_{n-1}^k,\quad n\neq k\\ \\ \\ C_{2}^1\cdot C_{4}^2 \cdot C_{6}^3\cdot ...\cdot C_{2n}^n = \\ \\ = \dfrac{2}{2-1}\cdot C_{1}^1\cdot \dfrac{4}{4-2}\cdot C_{3}^2\cdot \dfrac{6}{6-3}\cdot C_{5}^3\cdot ...\cdot \dfrac{2n}{2n-n}\cdot C_{2n-1}^{n} = \\ \\ = 2\cdot C_{1}^1\cdot 2\cdot C_{3}^2\cdot 2\cdot C_{5}^3\cdot ...\cdot 2\cdot C_{2n-1}^n =$

$= (2\cdot 2\cdot 2\cdot...\cdot 2)\cdot C_{1}^1\cdot C_{3}^2\cdot C_{5}^3\cdot...\cdot C_{2n-1}^n = \\ \\=2^n\cdot C_{1}^1\cdot C_{3}^2\cdot C_{5}^3\cdot...\cdot C_{2n-1}^n\quad (q.e.d)$