in Arithmetic, &c. the permutations or variations of any number of quantities; with regard to their position, order, &c. See COMBINATION.

To find all the possible CHANGES of any number of Quantities, or how oft their Order may be varied.] Suppose two quantities \(a\) and \(b\). Since they may be either wrote \(ab\) or \(ba\), it is evident their changes are \(2 = 2 \cdot 1\). Suppose three quantities \(abc\): their changes will be as in the margin; as is evident by combining \(c\) first with \(ab\), then with \(ba\); and hence \(acb\) the number of changes arises \(3 \cdot 2 \cdot 1 = 6\). If the quantities be 4, each may be combined four ways with each order of the other three; whence the number of changes arises \(6 \cdot 4 = 24\). Therefore, if the number of quantities be supposed \(n\), the number of changes will be \(n \cdot n - 1 \cdot n - 2 \cdot n - 3 \cdot n - 4\), &c. If the same quantity occur twice, the changes of two will be found \(bb\); of three, \(bab, abb, bba\); of four, \(eab, bca, bab, abc\). And thus the number of changes in the first case will be \(1 = (2 \cdot 1) : 2 \cdot 1\); in the second, \(3 = (3 \cdot 2 \cdot 1) : 2 \cdot 1\); in the third, \(12 = (4 \cdot 3 \cdot 2 \cdot 1) : 2 \cdot 1\).

If a fifth letter be added, in each series of four quantities, it will beget five changes, whence the number of all the changes will be \(60 = (5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)\). Hence if the number of quantities be \(n\), the number of changes will be \((n \cdot n - 1 \cdot n - 2 \cdot n - 3 \cdot n - 4\), &c.) : \(2 \cdot 2\).

From these special formulæ may be collected a general one, viz. if \(n\) be the number of quantities, and \(m\) the number which shows how oft the same quantity occurs; we shall have \((n \cdot n - 1 \cdot n - 2 \cdot n - 3 \cdot n - 4 \cdot n - 5 \cdot n - 6 \cdot n - 7 \cdot n - 8 \cdot n - 9\), &c.) : \((m - 1 \cdot m - 2 \cdot m - 3 \cdot m - 4\), &c.), the series being, to be continued, till the continual subtraction of unity from \(n\) and \(m\) leave 0. After the same manner we may proceed further, till putting \(n\) for the number of quantities, and \(l, m, r, &c.\) for the number that shows how oft any of them is repeated, we arrive at an universal form. \((n \cdot n - 1 \cdot n - 2 \cdot n - 3 \cdot n - 4 \cdot n - 5 \cdot n - 6 \cdot n - 7 \cdot n - 8\), &c.) : \((l \cdot l - 1 \cdot l - 2 \cdot l - 3 \cdot l - 4 \cdot l - 5\), &c. \(m \cdot m - 1 \cdot m - 2 \cdot m - 3 \cdot m - 4\), &c. \(r \cdot r - 1 \cdot r - 2 \cdot r - 3 \cdot r - 4\), &c.)

Suppose, for instance, \(n = 6, l = 3, r = 0\). The number of changes will be \((6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) : (3 \cdot 2 \cdot 1 \cdot 2 \cdot 1 \cdot 1) = (6 \cdot 5 \cdot 4) : (3 \cdot 2 \cdot 1 \cdot 2 \cdot 1 \cdot 1) = 240\).

Hence, suppose thirteen persons at a table, if it be required how oft they may change places; we shall find the number \(13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 6227020800\).

In this manner may all the possible anagrams of any word be found in all languages, and that without any study: suppose, v. g., it were required to find the anagrams of the word amor, the number of changes will be

\[ \begin{array}{cccc} a & o & a & m \\ o & a & m & r \\ m & o & r & a \\ r & a & m & o \\ \end{array} \]

The anagrams therefore of the word amor, in the Latin tongue, are roma, mora, maro, ramo, armo. See ANAGRAM.

Whether this new method of anagramatizing be like to prove of much service to that art, is left to the poets.