Sum with two powers

#sum#binomial#powers

Statement

Compute $\sum_{k=0}^{n}2^k3^{n-k}\binom{n}{k}$ .

Hint

Binomial theorem with $a=3$ , $b=2$ .

Show solutionHide solution

Binomial. $(3+2)^n=5^n=\sum_k\binom{n}{k}3^{n-k}2^k$ .

Conclusion. The sum equals $5^n$ .