combinations of m things out of n , for all possible values of m and n . Out of seven things one may be chosen in seven ways, and seven occurs in the eighth line of the second column. The combinations of two things chosen out of seven are 7 × 6/1 × 2 or 21, which is the third number in the eighth line. The combinations of three things out of seven are 7 × 6 × 5/1 × 2 × 3 or 35, which appears fourth in the eighth line. In a similar manner, in the fifth, sixth, seventh, and eighth columns of the eighth line I find it stated in how many ways I can select combinations of 4, 5, 6, and 7 things out of 7. Proceeding to the ninth line, I find in succession the number of ways in which I can select 1, 2, 3, 4, 5, 6, 7, and 8 things, out of 8 things. In general language, if I wish to know in how many ways m things can be selected in combinations out of n things, I must look in the n + 1 th line, and take the m + 1 th number, as the answer. In how many ways, for instance, can a subcommittee of five be chosen out of a committee of nine. The answer is 126, and is the sixth number in the tenth line; it will be found equal to 9 . 8 . 7 . 6 . 5/1 . 2 . 3 . 4 . 5, which our formula (p. 182) gives.
The full utility of the figurate numbers will be more apparent when we reach the subject of probabilities, but I may give an illustration or two in this place. In how many ways can we arrange four pennies as regards head and tail? The question amounts to asking in how many ways we can select 0, 1, 2, 3, or 4 heads, out of 4 heads, and the fifth line of the triangle gives us the complete answer, thus—
| We can select | No | head and 4 tails in 1 way. |
|---|---|---|
| " | 1 | head and 3 tails in 4 ways. |
| " | 2 | heads and 2 tails in 6 ways. |
| " | 3 | heads and 1 tail in 4 ways. |
| " | 4 | heads and 0 tail in 1 way. |
The total number of different cases is 16, or 24, and when we come to the next chapter, it will be found that these numbers give us the respective probabilities of all throws with four pennies.
I gave in p. 181 a calculation of the number of ways in which eight planets can meet in conjunction; the reader will find all the numbers detailed in the ninth line of the arithmetical triangle. The sum of the whole line is 28 or 256; but we must subtract a unit for the case where no planet appears, and 8 for the 8 cases in which only one planet appears; so that the total number of conjunctions is 28 – 1 – 8 or 247. If an organ has eleven stops we find in the twelfth line the numbers of ways in which we can draw them, 1, 2, 3, or more at a time. Thus there are 462 ways of drawing five stops at once, and as many of drawing six stops. The total number of ways of varying the sound is 2048, including the single case in which no stop at all is drawn.
One of the most important scientific uses of the arithmetical triangle consists in the information which it gives concerning the comparative frequency of divergencies from an average. Suppose, for the sake of argument, that all persons were naturally of the equal stature of five feet, but enjoyed during youth seven independent chances of growing one inch in addition. Of these seven chances, one, two, three, or more, may happen favourably to any individual; but, as it does not matter what the chances are, so that the inch is gained, the question really turns upon the number of combinations of 0, 1, 2, 3, &c., things out of seven. Hence the eighth line of the triangle gives us a complete answer to the question, as follows:—
Out of every 128 people—