By John E. Maxfield

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**Sample text**

Since S has a finite number of elements, we can list them in some order as How many different second rows, and therefore how many different permu tations, can we form ? A permutation can take sl9 say, into any of the n elements of S. (Can Si be mapped to ? ) Having chosen an image tx for sx from S, we have only n — possible images t2 for s2, because no two elements have the same image under a permutation, so h ** h. Then with two elements used already as images, there are only n — 2 choices for r3, and so forth.

4) Sloat are to be seated along one side of a speaker’s table. Determine by reference to Exercise 3-4 how many seating arrangements are possible for the four people. Write down (code 1, 2, 3, 4) the seating arrangements that alternate men and women. What is the probability that a random permutation of the four people will alternate men and women? Theorem 3-2. under composition. Q)P. ) Because Q is a permutation, there is exactly one image sQ in S , which then has exactly one Let P be 2 3 4\ 3 4 1/ and Q be The composite mapping P o Q maps any element into exactly one element of S.

A classic illustration of the peculiarities of arithmetic in such a case is that of the hotel with rooms numbered 1, 2, 3, . . , one room for each natural number. Suppose every room is full, but another guest arrives. The manager simply gives him room No. 1, whose former occupant moves to No. 2, whose former occupant moves to No. 3, and so on. What would be an im possible problem in a finite hotel is solved easily in a countably infinite hotel. Definition 2-10. A set is called countably infinite if its members can be put into one-to-one correspondence with the natural numbers.