PERMUTATIONS AND COMBINATIONS. 321
which we can fill r places when we have n different things at our disposal.
The first place may be filled in n ways, for any one of the n things may be taken; when it has been filled in any one of these ways, the second place can then be filled in n-1 ways; and since each way of filling the first place can be associated with each way of filling the second, the number of ways in which the first two places can be filled is given by the product n(n-1). And when the first two places have been filled in any way, the third place can be filled in n-2 ways. And reasoning as before, the number of ways in which three places can be filled is n(n-1)(n-2).
Proceeding thus, and noticing that a new factor is introduced with each new place filled, and that at any stage the number of factors is the same as the number of places filled, we shall have the number of ways in which r places can be filled equal to
n(n - 1)(n - 2) ... to r factors.
We here see that each factor is formed by taking from n a number one less than that which applies to the place filled by that factor; hence the rth factor is n-(r-1), or n-r+1.
Therefore the number of permutations of n things taken r at a time is
n(n - 1)(n - 2)---(n - r +1).
Cor. The number of permutations of n things taken all at a time is
n(n -1)(n - 2) ... to n factors, or n(n-1)(n - 2) ... 3.2.1.
It is usual to denote this product by the symbol n, which is read “factorial n.” Also n! is sometimes used for n.
393. We shall in future denote the number of permutations of n things taken r at a time by the symbol nPr, so that
nPr = n(n-1)(n-2)---(n-r+1); also nPr = n.
