COMBINATION, doctrine of Prob. 1. Any number of quantities being given, together with the number in each com bination, to find the number of combina tions. One quantity admits of no combi nation : two, a and b, only of one combi nation ; of three quantities, a b c, there are three combinations, viz. a b, a c, b c; of four quantities, there are six combina tions, viz, a b, a c, a d, b c,b d, c d ; of five quantities, there are ten combinations, viz. a b, a c, b c, a d, b (I, c d, a b e, c e, d e. Hence it appears that the numbers of combinations proceed as 1, 3, 6, 10 ; that is, they are triangular numbers, whose sides differ by unity from the number of given quantities. If this then be supposed .7, the side of the number of combinations will be q — 1, and so the number of combinations q 1, g + 0.
2 See TRIANGULAR NUMBERS.
If three quantities are to be combined, and the number in each combination be three, there will be only one combina tion, a b c; if a fourth be added, four combinations will be found, a b c, a b d, b c d, a c d; if a fifth be added, the com binations will be ten, viz. a b c, a b d, b c d,
a c d, a b e, b d e, b c e, ace, a d e ; if a sixth, the combinations will be twenty, &c. The numbers, therefore, of combi nations proceed as 1, 4, 10, 20, &c. that is, they are the first pyramidal triangular numbers, whose side differs by two units from the number of given quantities. Hence, if the number of given quantities beg, the side will be q— 2, and so the number of combinations 9 — 1 2 9+0 , 3 If four quantities are to be combined, we shall find the numbers of combina tions to proceed as pyramidal, triangular numbers of the second order, 1, 5, 15, 35, &c. whose side differs from the num ber of quantities by the exponent minus an unit. Wherefore if the number of quantities be q, the side will be q — 3, and the number of combinations 32 1 ry — 2, g —1 q +0.
I See PYRAMIDAL 2 3 4