PERMUTATIONS AND COMBINATIONS (Lat. permutatio, from permutare, to change en tirely, from per, through + mature, to change, frequentative of movere, to move, Skt, uric, to push). The different groups of r things which can be selected from a collection of n different things. without reference to their arrangement, are called the combinations of tt things taken r at a time. For example, the combinations of the four letters a, b. c, d, taken 3 at a time, are vibe, abd, acct, bed; taken 2 at a time, ab, rte. ad, be, bd. ed. The different groups of r things which can be selected from n different things, varying the arrangement in every possible manner, are called the permutations of n things taken r at a time. E.g. the permutations of the letters a, b, c, taken 2 at a time are ab, ba, cc, ea. be, cb. The number of combinations of n things taken r at a time is indicated by the symbol The number of permutations of n things taken rat a time is indicated by the symbol PD, The chief properties of permutations and combinations are: ( I) The number of permutations of a different things taken r at a time is n(a-1) (n-2) .... (a—r+1). E.g. the number of permutations of the letters of the word courage taken three at a time is 7 -6.5 = 210. ( 2) The number of permuta tions of a. things taken all together is a (n-1) 3.2I = a! (3) The number of mutations of a different things taken r at a time, when each of the a things may be repeated, is nr; e.g. the number of ways of selecting 3 numbers front 50 on a combination lock, repetitions being allowed. is 50' = 125.000. (4) The number of
combinations of a different things taken rat a time is n (n— (n— 2) (n— r 1 ) r ! n! or r! (n—r)1 e.g. 3 persons can he selected from a class of 20 20 • 19 • 18 in = 1140 different ways. 3! The formulas of permutations and combina tions express many relations of both algebra and geometry and possess a peculiar interest in math ematics. E.g. the coefficients in the binomial expansion for a positive integral exponent may be expressed by formulas of combinations thus, (a ± = a° + ± ''',, I '1 The maximum number of vertices of a general polygon of a sides is expressed by C. Also, such problems as those of combination locks, of the number of signals with a given system of signs, and of forming all possible numbers Iron' given digits, arc solved with unusual brevity by means (4 the formulas of permutation and combination. This subject was known to the Dindus, partien L(.rly to Bhaskara (1,1114), and is related to the subject of probability (q.v.). Its principles are often explained in text-books under the title Choice. By cyclic permutation is meant the in terchange of the elements of a function in cyclic order. E.g. (a—b)±(b—c)-1- (c—a) becomes (b—c)+(c—a)-1- (a—b) by the cyclic inter change of a for b, b for c, and c for a.