\begin{aligned} ans & = \sum_{i=0}^n\dbinom{n}{i}p^iq^{n-i}i^k \\ & =\sum_{i=0}^n {n\choose i}p^iq^{n-i}\sum_{j=0}^k{k \brace j}i^{\underline{j} } \\ & =\sum_{i=0}^n\sum_{j=0}^k {k \brace j}{n\choose i}{i\choose j}p^iq^{n-i} j!\\ & =\sum_{i=0}^n\sum_{j=0}^k {k \brace j}{n\choose j}{n-j\choose i-j}p^iq^{n-i} j!\\ & =\sum_{j=0}^k{k \brace j}{n\choose j}j!p^j \sum_{i=0}^n {n-j\choose i-j}p^{i-j}q^{n-i} \\ & =\sum_{j=0}^k{k \brace j}n^{\underline{j} }p^j\sum_{i=0}^{n-j} {n-j\choose i}p^iq^{n-j-i}\\ & =\sum_{j=0}^k{k \brace j}n^{\underline{j} }p^j(p+q)^{n-j} \\ \end{aligned}