COMBINATIONS
, denote the alternations or variations of any number of quantities, letters, sounds, or the like, in all possible ways.
Father Mersenne gives the combinations of all the notes and sounds in music, as far as 64; the sum of which amounts to a number expressed by 90 places of figures. And the number of possible combinations of the 24 letters of the alphabet, taken first two by two, then three by three, and so on, according to Prestet's calculation, amounts to
Father Truchet, in Mem. de l'Acad. shews, that two square pieces, each divided diagonally into two colours, may be arranged and combined 64 different ways, so as to form so many different kinds of chequer-work: a thing that may be of use to masons, paviours, &c.
Doctrine of Combinations.
I. Having given any number of things, with the number in each combination; to find the number of combinations.
One thing admits of no combination.
Two, a and b, admit of one only, viz ab.
Three, a, b, c, admit of three, viz ab, ac, bc.
Four admit of six, viz, ab, ac, ad, bc, bd, cd.
Five admit of 10, viz, ab, ac, ad, ae, bc, bd, be, cd, ce, de.
Whence it appears that the numbers of combinations, of two and two only, proceed according to the triangular numbers 1, 3, 6, 10, 15, 21, &c, which are produced by the continual addition of the ordinal series 0, 1, 2, 3, 4, 5, &c. And if n be the number of things, then the general formula for expressing the sum of all their combinations by twos, will be .
| Thus, if n = 2; this becomes 2.1/2 | = 1. |
| If n = 3; it is 3.2/2 " | =3. |
| If n = 4; it is 4.3/2 " | = 6. &c. |
Three things admit of one order, abc.
Four admit of 4; viz abc, abd, acd, bcd.
Five admit of 10; viz abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde. And so on according to the first pyramidal numbers 1, 4, 10, 20, &c, which are formed by the continual addition of the former, or triangular numbers 1, 3, 6, 10, &c. And the general formula for any number n of combinations, taken by threes, is .
So, if . &c. Proceeding thus, it is found that a general formula for any number n of things, combined by m at each time, is , continued to m factors, or terms, or till the last factor in the denominator be m.
So, in 6 things, combined by 4's, the number of combinations is .
3. By adding all these series together, their sum will be the whole number of possible combinations of n things combined both by twos, by threes, by fours, &c. And as the said series are evidently the coefficients of the power n of a binomial, wanting only the first two 1 and n; therefore the said sum, or whole number of all such combinations, will be ―(1 + 1) - n - 1, or 2n - n - 1. Thus if the number of things be 5; then .
II. To find the number of Changes and Alterations which any number of quantities can undergo, when combined in all possible varieties of ways, with themselves and each other, both as to the things themselves, and the Order or Position of them.
One thing admits but of one order or position.
Two things may be varied four ways; thus, aa, ab, ba, bb.
Three quantities, taken by twos, may be varied nine ways; thus aa, ab, ac, ba, ca, bb, bc, cb, cc.
In like manner four things, taken by twos, may be varied 42 or 16 ways; and 5 things, by twos, 52 or 25 ways; and, in general, n things, taken by twos, may be changed or varied n2 different ways.
For the same reason, when taken by threes, the changes will be n3; and when taken by fours, they will be n4; and so generally, when taken by n's, the changes will be nn.
Hence, then, adding all these together, the whole number of changes, or combinations in n things, taken both by 2's, by 3's, by 4's, &c, to n's, will be the sum of the geometrical series n + n2 + n3 + n4 nn, which sum is .
For example, if the number of things n be 4; this gives .
And if n be 24, the number of letters in the alphabet; the theorem gives . In so many different ways, therefore, may the 24 letters of the alphabet be varied or combined among themselves, or so many different words may be made out of them.
