| x ( x - 1)( x - 2) | ; |
|---|---|
| 1 × 2 × 3 |
that of four is
| x ( x - 1)( x - 2)( x - 3) | ; |
|---|---|
| 1 × 2 × 3 × 4 |
- 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 |
in which the product 7 × 6 × 5 × 4 occurs in both terms, and therefore may be removed from both (108), leaving
| 10 × 9 × 8 | , |
|---|---|
| 1 × 2 × 3 |
which is the number of combinations of three out of ten. The same may be shewn in other cases.
EXERCISES.
How many combinations of four can be made out of twelve things?
Answer, 495.
