Chapter 1, Problem 1.20STPE

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

a. by a combinatorial argument that first notes that $r^n$ is the number of different n letter sequences that can be formed from an alphabet consisting of r letters, and then determines how many of these letter sequences have letter 1 a total of $x_1$ times and letter 2 a total of $x_2$ times and ... and letter r a total of $x_r$ times:

b. by using the multinomial theorem.