COMBINATIONS AND PERMUTA-• TIONS. Combinations and permutations deal with •the arrangement and grouping of ob jects. They arose out of problems such as these: °How many numbers can be thrown with two dice?° °How many striped flags can be made with three colors?° °How many whist hands of 13 cards each can be dealt from a pack of 52 cards, the cards being put back and shuffled after each deal?)) Although games of chance furnished most of the problems in the begin ning, the modern aspect of the subject finds ap plications in statistics, probabilities and the the ories of numbers, finite group, and algebraic forms. Aside from isolated references to sin gle questions in the works of the Greek mathe maticians, the earliest attempt at organized treatment of combinations and permutations occurs in the treatise of the Hindu astronomer Bhaskara Acharya in the 12th century. The subject received no further attention until the time of Cardan (1501-76) and another cen tury elapsed before Pascal (1623-62) succeeded in building it into a science, as yet, however, in its infancy. Hindenburg (1741-1808) is re garded as the founder of the modern theory, but his school, known as the °combinatory school° held aloof from the rest of mathemat ics. Since theri the researches of Jacobi, De Morgan, Sylvester, Cayley, Franklin, Mathews and MacMahon have shown its use in the gen eral theories of arithmetic and algebra. We
shall take up only elementary combinations and permutations; for a complete treatment of com binatory analysis consult MacMahon,
Permutations.— A permutation of n ele ments taken r at a time is a linear arrange ment, as in a row, of r of those n things with regard to the order in which they may be placed; e.g., a b c and a c b are permutations of a, b, and c. This name was given by Jacob Bernoulli ((Ars conjectandi) 1713). The num ber of permutations will be denoted by Pn, r, n being the total number of elements and r the number in a group; the notation (n r) or (n), is often used. To find Pn, r, observe that the first one of the r objects may be selected in n ways and for each selection there are n-1 things left from which to choose the second object; both may thus be chosen in ways. By continuing the argument we find: Pn, r (n--2).... there being r factors. The following table gives numerical values of Pn, r from n=4 to 10 and r=2 to 10. From the table Ps, I, =6720 and is found under n=8 and opposite r=5.