The exact probabilities of equation (9-9) rest on the claim that the number of ways of add...

The exact probabilities of equation (9-9) rest on the claim that the number of ways of adding N distinct non-negative integers to give a total of M is (M + N - 1)!/ [M!(N - 1)!]. One way to prove it involves the following trick. It represents two ways that N distinct integers can add to M—9 and 5, respectively, in this special case.

The X's represent the total of the integers, M—each row has 5. The I's represent "dividers" between the distinct integers, of which there will of course be N — 1—each row has 8. The first row says that n1 is 3 (three X's before the divider between it and n2), n2 is 0 (no X's between its left divider with n1 and its right divider with n3) is 1, n4 through n6 are 0, n7 is 1, and rig and n9 are 0. The second row says that n2 is 2, n6 is 1, n9 is 2, and all other n are 0. Further rows could account for all possible ways that the integers can add to M. Argue that, properly applied, the binomial coefficient (discussed in Appendix J) can be invoked to give the correct total number of ways for any N and M.