(2.17) |

So, , and , and , and so on. Clearly, the number of ways of potting six distinguishable pool balls into six pockets is (which incidentally equals 720). Because there is nothing special about pool balls, or the number six, we can safely infer that the number of different ways of arranging distinguishable objects, denoted , is given by

(2.18) |

Suppose that we take the number four ball off the pool table, and replace it by a second number five ball. How many different ways are there of potting the balls now? Consider a previous arrangement in which the number five ball was potted into the top-left pocket, and the number four ball was potted into the top-right pocket, and then consider a second arrangement that only differs from the first because the number four and five balls have been swapped around. These arrangements are now indistinguishable, and are therefore counted as a single arrangement, whereas previously they were counted as two separate arrangements. Clearly, the previous arrangements can be divided into two groups, containing equal numbers of arrangements, that differ only by the permutation of the number four and five balls. Because these balls are now indistinguishable, we conclude that there are only half as many different arrangements as there were before. If we take the number three ball off the table, and replace it by a third number five ball, then we can split the original arrangements into six equal groups of arrangements that differ only by the permutation of the number three, four, and five balls. There are six groups because there are separate permutations of these three balls. Because the number three, four, and five balls are now indistinguishable, we conclude that there are only the number of original arrangements. Generalizing this result, we conclude that the number of arrangements of indistinguishable and distinguishable objects is

(2.19) |

We can see that if all the balls on the table are replaced by number five balls then there is only possible arrangement. This corresponds, of course, to a number five ball in each pocket. A further straightforward generalization tells us that the number of arrangements of two groups of and indistinguishable objects is