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820 ALGEBRA. 391. Fundamental Principle. Before discussing the general propositions of this chapter the following important principle should be carefully noticed.

If one operation can he performed in m ways, and (when it has been performed in any one of these ways) a second operation can then he performed in n ways; the number of ways of performing the two operations will be m x n. If the first operation be performed in any one way, we can associate with this any of the n ways of performing the second operation; and thus we shall have n ways of performing the two operations without considering more than one way of performing the first; and so, corresponding to each of the m ways of performing the first operation, we shall have n ways of performing the two; hence the product m X n represents the total number of ways in which the two operations can be performed.

Ex. Suppose there are 10 steamers plying between New York and Liverpool: in how many ways can a man go from New York to Liverpool and return by a different steamer?

There are ten ways of making the first passage; and with each of these there is a choice of nine ways of returning (since the man is not to come back by the same steamer); hence the number of ways of making the two journeys is 10 x 9, or 90.

This principle may easily be extended to the case in which there are more than two operations each of which can be performed in a given number of ways.

Ex. Three travellers arrive at a town where there are four hotels; in how many ways can they take up their quarters, each at a different hotel?

The first traveller has choice of four hotels, and when he has made his selection in any one way, the second traveller has a choice of three; therefore the first two can make their choice in 4 x 3 ways; and with any one such choice the third traveller can select his hotel in 2 ways; hence the required number of ways is 4 x 3 x 2, or 24.

392. To find the number of permutations of n dissimilar things taken r at a time.

This is the same thing as finding the number of ways in