326 ALGEBRA.
group to consist of five dissimilar things and therefore capable of 5 arrangements among themselves. [Art. 392, Cor.]
398. To find the permutations of n things, taking them all at a time, when p things are of one kind, q of another kind, r of a third kind, and the rest all different.
Let there be n letters; suppose p of them to be a, q of them to be b, r of them to be c, and the rest to be unlike.
Let x be the required number of permutations; then if in any one of these permutations the p letters a were replaced by p unlike letters different from any of the rest, from this single permutation, without altering the position of any of the remaining letters, we could form p new permutations. Hence if this change were made in each of the x permutations, we should obtain a x p permutations.
Similarly, if the q letters b were replaced by q unlike letters, the number of permutations would be x x p × q.
In like manner, by replacing the r letters c by r unlike letters, we should finally obtain x x p × q × r permutations.
But the things are now all different, and therefore admit of n permutations among themselves. Hence x x p × q × r = n;
that is, x = n p q r;
which is the required number of permutations.
Any case in which the things are not all different may be treated similarly.
Ex. 1. How many different permutations can be made out of the letters of the word assassination taken all together?
We have here 13 letters of which 4 are s, 3 are a, 2 are i, and 2 are n. Hence the number of permutations
= 13 4 3 2 2 = 13.11.10.9.8.7.3.5 = 1001 x 10800 = 10810800.
