$$\sum_{x=1}^{n}{x^2 \binom{n}{x} p^x (1-p)^{n-x} }$$
You asked:
Evaluate the expression: \(\sum_{x = 1}^{n} {{x}^{2} \cdot \operatorname{C}(n, x) \cdot {p}^{{\left( x \cdot \left(1 - p\right) \right)}^{n - x}}}\)
MathBot Answer:
\[\sum_{x=1}^{n} p^{\left(x \left(1 - p\right)\right)^{n - x}} x^{2} \operatorname{C}(n, x) = \sum_{x=1}^{n} p^{\left(x \left(1 - p\right)\right)^{- x} \left(x \left(1 - p\right)\right)^{n}} x^{2} {\binom{n}{x}}\]