Why is a portion of the expected value for a binomial random variable equal to 1?

Question

Trying to follow a proof for the expected value of a binomial random variable equal to np. I'm stumped on this step:

why does $$\sum_{j=0}^m\binom{m}{j}p^j(1-p)^{m-j}$$ result in 1?

Answer 1

Thanks to @Rumpelstiltskin, I figured out that, according to the binomial theorem,

$$(a + b)^n = \sum_{k=0}^n\binom{n}{k}a^{n-k}b^k$$

if we let a = p, b = 1 - p, n = m, and k = j

$$\sum_{j=0}^m\binom{m}{j}p^j(1-p)^{m-j} = (p + (1 - p))^m = 1$$