Also, since any combination of four, abcd, contains 4 × 3 × 2 × 1 permutations, the number of combinations of four is
| 7 × 6 × 5 × 4 | , |
| 1 × 2 × 3 × 4 |
and so on. The rule is: To find the number of combinations, each containing n counters, divide the corresponding number of permutations by the product of 1, 2, 3, &c. up to n. If x be the whole number, the number of combinations of two is
| x(x - 1) | ; |
| 1 × 2 |
that of three is
| x(x - 1)(x - 2) | ; |
| 1 × 2 × 3 |
that of four is
| x(x - 1)(x - 2)(x - 3) | ; |
| 1 × 2 × 3 × 4 |
211. The rule may in half the cases be simplified, as follows. Out of ten counters, for every distinct selection of seven which is taken, a distinct combination of 3 is left. Hence, the number of combinations of seven is as many as that of three. We may, therefore, find the combinations of three instead of those of seven; and we must moreover expect, and may even assert, that the two formulæ for finding these two numbers of combinations are the same in result, though different in form. And so it proves; for the number of combinations of seven out of ten is
| 10 × 9 × 8 × 7 × 6 × 5 × 4 | , |
| 1 × 2 × 3 × 4 × 5 × 6 × 7 |