Though the properties above explained are highly curious, the greatest value of the triangle arises from the fact that it contains a complete statement of the values of the formula (p. 182), for the numbers of 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 2^{4}, 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.