Thus if we were to permutate the letters of the name Ann , according to our previous rule, we should obtain 3 × 2 × 1, or 6 orders; but half of these arrangements would be identical with the other half, because the interchange of the two n ’s has no effect. The really different orders will therefore be 3 . 2 . 1/1 . 2 or 3, namely Ann , Nan , Nna . In the word utility there are two i ’s and two t ’s, in respect of both of which pairs the numbers of permutations must be halved. Thus we obtain 7 . 6 . 5 . 4 . 3 . 2 . 1/1 . 2 . 1 . 2 or 1260, as the number of permutations. The simple rule evidently is—when some things or letters are undistinguished, proceed in the first place to calculate all the possible permutations as if all were different, and then divide by the numbers of possible permutations of those series of things which are not distinguished, and of which the permutations have therefore been counted in excess. Thus since the word Utilitarianism contains fourteen letters, of which four are i ’s, two a ’s, and two t ’s, the number of distinct arrangements will be found by dividing the factorial of 14, by the factorials of 4, 2, and 2, the result being 908,107,200. From the letters of the word Mississippi we can get in like manner 11!/4! × 4! × 2! or 34,650 permutations, which is not the one-thousandth part of what we should obtain were all the letters different.
Calculation of Number of Combinations.
Although in many questions both of art and science we need to calculate the number of permutations on account of their own interest, it far more frequently happens in scientific subjects that they possess but an indirect interest. As I have already pointed out, we almost always deal in the logical and mathematical sciences with combinations, and variety of order enters only through the inherent imperfections of our symbols and modes of calculation. Signs must be used in some order, and we must withdraw our attention from this order before the signs correctly represent the relations of things which exist neither before nor after each other. Now, it often happens that we cannot choose all the combinations of things, without first choosing them subject to the accidental variety of order, and we must then divide by the number of possible variations of order, that we may get to the true number of pure combinations.
Suppose that we wish to determine the number of ways in which we can select a group of three letters