Combinatorics and Graph Theory · Problem 1.3.11

Fundamentals — David Guichard

Mathematics — Combinatorics · binomial-theorem

Problem

Use

(x+1)^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i} x^i

to find a simple expression for

\displaystyle\sum_{i=1}^{n} i\binom{n}{i} x^{i-1}

. Then find a simple expression for

\displaystyle\sum_{i=1}^{n} i\binom{n}{i}