Question

Alguém pode me ajudar ?Prove que:

Answer 1

Resposta:

$Desenvolvimento \ de \ um \ binomio \ de \ Newton:\\(x + y)^n = \binom{n}{0}x^{0}.y^{n-0} + \binom{n}{1}x^1.y^{n-1}... \binom{n}{n-1}x^{n-1}.y^{n-(n-1)} + \binom{n}{n}x^{n}.y^{n-n}\\\\Se \ fizermos \ x=y=1 \ fica:\\\\(1 + 1)^n = \binom{n}{0}1^{0}.1^{n-0} + \binom{n}{1}1^1.1^{n-1}... \binom{n}{n-1}1^{n-1}.1^{n-(n-1)} + \binom{n}{n}1^{n}.1^{n-n}\\(1 + 1)^n = \binom{n}{0} + \binom{n}{1}... \binom{n}{n-1} + \binom{n}{n}\\\\2^n = \binom{n}{0} + \binom{n}{1}... \binom{n}{n-1} + \binom{n}{n}$

$\bold{2^n = C^0_n + C^1_n + ... C^{n-1}_n + C^n_n}$