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PERMUTATIONS AND COMBINATIONS. 327 Ex. 2. How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1, so that the odd digits always occupy the odd places?

The odd digits 1, 3, 3, 1 can be arranged in their four places in

4 2 2 ways (1).

The even digits 2, 4, 2 can be arranged in their three places in

3 12 ways (2).

Each of the ways in (1) can be associated with each of the ways in (2).

Hence the required number = 4 2 2 \times 3 2 = 6 x 3 = 18.

399. To find the number of permutations of n things r at a time, when each thing may be repeated once, twice, ... up to r times in any arrangement.

Here we have to consider the number of ways in which r places can be filled when we have n different things at our disposal, each of the n things being used as often as we please in any arrangement.

The first place may be filled in n ways, and, when it has been filled in any one way, the second place may also be filled in n ways, since we are not precluded from using the same thing again. Therefore the number of ways in which the first two places can be filled is n \times n or n2.

The third place can also be filled in n ways, and therefore the first three places in n3 ways.

Proceeding thus, and noticing that at any stage the index of n is always the same as the number of places filled, we shall have the number of ways in which the r places can be filled equal to nr.

Ex. In how many ways can 5 prizes be given away to 4 boys, when each boy is eligible for all the prizes?

Any one of the prizes can be given in 4 ways; and then any one of the remaining prizes can also be given in 4 ways, since it nay be obtained by the boy who has already received a prize. Thus two prizes can be given away in 42 ways, three prizes in 43 ways, and so on. Hence the 5 prizes can be given away in 45, or 1024 ways.