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 CHAPTER XXXV.

Permutations and Combinations.

390. Each of the arrangements which can be made by taking some or all of a number of things is called a permutation.

Each of the groups or selections which can be made by taking some or all of a number of things is called a combination.

Thus the permutations which can be made by taking the letters a, b, c, d two at a time are twelve in number; namely,

ab, ac, ad, bc, bd, cd, ba, ca, da, cb, db, dc; each of these presenting a different arrangement of two letters.

The combinations which can be made by taking the letters a, b, c, d two at a time are six in number; namely,

ab, ac, ad, be, bd, cd;

each of these presenting a different selection of two letters.

From this it appears that in forming combinations we are only concerned with the number of things each selection contains; whereas in forming permutations we have also to consider the order of the things which make up each arrangement; for instance, if from four letters a, b, c, d we make a selection of three, such as abc, this single combination admits of being arranged in the following ways:

abc, acb, bca, bac, cab, cba,

and so gives rise to six different permutations.