Question

Să se calculeze suma: ex A9 -

subpunctul a)

Clasa a 10-a , combinări

Answer 1

$S_n = C_{n}^1+2C_{n}^2+3C_{n}^3+...+nC_n^n \\ \\ kC_{n}^k = k\dfrac{n!}{k!(n-k)!} = \dfrac{n!}{(k-1)!(n-k)!} = \\ \\ = \dfrac{(n-1)!\,n}{(k-1)!\Big((n-1)-(k-1)\Big)!} = nC_{n-1}^{k-1} \\ \\ S_n = nC_{n-1}^0+nC_{n-1}^1+nC_{n-1}^{2}+...+nC_{n-1}^{n-1} \\ \\ S_n = n\Big(C_{n-1}^0+C_{n-1}^1+C_{n-1}^2+...+C_{n-1}^{n-1}\Big)\\ \\ \boxed{S_n = n\cdot 2^{n-1}}\\ \\\\ (1+x)^n = C_{n}^0+C_{n}^1x+C_{n}^2x^2+...+C_{n}^nx^n$

$(1+1)^n = C_{n}^0+C_{n}^1+C_{n}^2+...+C_{n}^n \\ \\ \Rightarrow C_{n}^0+C_{n}^1+C_{n}^2+...+C_{n}^n = 2^n \\ \\\Rightarrow C_{n-1}^0+C_{n-1}^1+C_{n-1}^2+...+C_{n-1}^{n-1} = 2^{n-1}$