Chapter 15.4, Problem 21P

The following problem arises in quantum mechanics (see Chapter 13, Problem 7.21 ). Find the number of ordered triples of nonnegative integers a,b,c whose sum $a+b+c$ is a given positive integer n. (For example, if n=2, we could have $(a,b,c)=(2,0,0)$ or $(0,2,0)\text{ or }(0,0,2)\text{ or }(0,1,1)\text{ or }(1,0,1)\text{ or }(1,1,0).)$ Hint: Show that this is the same as the number of distinguishable distributions of $n$ identical balls in 3 boxes, and follow the method of the diagram in Example 5.