Mathematics, are the different collections or groups which may be formed out of a given number of things, taking a certain number at a time, without any regard to the order in which they may follow each other.

Combinations are thus distinguished from permutations, a term used to express the different ways in which the things which constitute a combination may be arranged. For example, let the things to be combined be the four letters \(a, b, c, d\). These may evidently form six combinations taken two by two, viz.

\[ ab, ac, ad, bc, bd, cd. \]

Taken three by three, four combinations may be formed, viz.

\[ abc, abd, acd, bcd. \]

There is only the single combination abed of all the four. There may, however, be twenty-four permutations of the four letters, and of these, six begin with \(a\), viz.

\[ abcd, abdc, acbd, acdb, adbc, adcb. \]

It is evident that a like number may begin with each of the remaining letters.

The doctrine of combinations, like every other mathematical speculation, may be considerably extended. The most interesting problem, however, is this.

Problem.—A number of things denoted by the letters \(a, b, c, d, \&c.\) being given; to find in how many ways they may be combined two by two, three by three, four by four, &c.

Solution.—Let \(n\) denote the number of things. The number of ways they can be combined

two by two is \(\frac{n(n-1)}{1 \cdot 2}\),

three by three, \(\frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3}\),

four by four \(\frac{n(n-1)(n-2)(n-3)}{1 \cdot 2 \cdot 3 \cdot 4}\);

and in general the number of combinations, taking them \(m\) by \(m\), is

\[ \frac{n(n-1)(n-2)(n-3)\ldots(n-m+1)}{1 \cdot 2 \cdot 3 \cdot \ldots \cdot m}. \]

Suppose, for example, it were required to find how many combinations may be formed out of thirteen cards, all different from each other, taken four at a time.

Here we have \(\frac{13 \cdot 12 \cdot 11 \cdot 10}{1 \cdot 2 \cdot 3 \cdot 4} = 715\) combinations.

It is observable that the expressions

\[ \frac{n(n-1)}{1 \cdot 2}, \quad \frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3}, \] which denote the number of times the things can be taken, one by one, two by two, three by three, &c. are the coefficients of the second, third, fourth, &c. terms of a binomial raised to the nth power (Algebra, § 160). Therefore, their sum increased by an unit will be \((1 + 1)^n = 2^n\), and hence the number of all the possible combinations of \(n\) things is \(2^n - 1\).

The theory of combinations is of great use in the doctrine of chances or probabilities; for the numerical measure of the probability of the occurrence of an event is a fraction whose numerator is the number of cases in which the event may happen, and denominator the number in which it may both happen and fail; and these are determined by the doctrine of combinations. Thus, if it were required to find the chance of drawing any four specified cards out of a pack of fifty-two, we must find how many combinations of four may be formed out of fifty-two things. The number, by the formula, will be

\[ \frac{52 \cdot 51 \cdot 50 \cdot 49}{1 \cdot 2 \cdot 3 \cdot 4} = 270725; \]

and since each is as likely to happen as any one of the remainder, the probability of drawing the four specified cards will be measured by the fraction \(\frac{1}{270725}\).

The following works may be consulted on the theory of combinations:—James Bernoulli Ars Conjectandi, cap. 2. This was translated and published by the late Baron Masses, in a volume of tracts entitled The Doctrine of Permutations and Combinations, &c.; Dr Wallis's Algebra; and most treatises of algebra, among the latest of which is Peacock's, Cambridge, 1830.