Variance of Binomial Distribution/Proof 2

Proof
From Variance of Discrete Random Variable from PGF:
 * $\operatorname{var} \left({X}\right) = \Pi''_X \left({1}\right) + \mu - \mu^2$

where $\mu = E \left({x}\right)$ is the expectation of $X$.

From the Probability Generating Function of Binomial Distribution:
 * $\Pi_X \left({s}\right) = \left({q + p s}\right)^n$

where $q = 1 - p$.

From Expectation of Binomial Distribution:
 * $\mu = n p$

From Derivatives of PGF of Binomial Distribution:
 * $\Pi''_X \left({s}\right) = n \left({n - 1}\right) p^2 \left({q + p s}\right)^{n - 2}$

Setting $s = 1$ and using the formula $\Pi''_X \left({1}\right) + \mu - \mu^2$:
 * $\operatorname{var} \left({X}\right) = n \left({n - 1}\right) p^2 + n p - n^2 p^2$

Hence the result.