Chapter 1, Problem 1.20STPE

Verify that the equality ${\displaystyle \underset{x_1+\mathrm{...}+x_r=n,x_1\ge 0}{\sum }\frac{n!}{x_1!x_2!\mathrm{...}{x}_r!}}=r^n$

when $n=3,r=2$, and then show that it always valid. (The sum is over all vectors of r nonnegative integer values whose sum is n.) Hint: How many different n letter sequences can be formed from the first r letters of the alphabet? How many of them use letter i of the alphabet a total of $x_i$ times for each $i=1,\mathrm{...}r$?