Page:Elementary algebra (1896).djvu/340

322 ALGEBRA. In working numerical examples it is useful to notice that the suffix in the symbol nPr, always denotes the number of factors in the formula we are using.

Ex. 1. Four persons enter a carriage in which there are six seats: in how many ways can they take their places?

The first person may seat himself in 6 ways; and then the second person in 5; the third in 4; and the fourth in 3; and since each of these ways may be associated with each of the others, the required answer is 6 x 5 x 4 x 3, or 360.

Ex. 2. How many different numbers can be formed by using six out of the nine digits 1, 2, 3, ... 9?

Here we have 9 different things, and we have to find the number of permutations of them taken 6 at a time;

the required result = 9P6 =9x8x7x6x5x 4= 60480.

394. To find the number of combinations of n dissimilar things taken r at a time.

Let nCr, denote the required number of combinations.

Then each of these combinations consists of a group of r dissimilar things which can be arranged among themselves in r ways. [Art. 392, Cor.]

Hence nCr, x r is equal to the number of arrangements of n things taken r at a time; that is,

nCr r =nPr = n(n - 1(n-2)...(n- r +1); nCr = n(n -1)(n - 2)...(n-r+1) r (1)

Cor. This formula for nCr may also be written in a different form; for if we multiply the numerator and the denominator by n-r we obtain

n(n - 1)(n - 2)...(n-r+1) n-r r n-r or,  n r n-r, (2)

since n(n - 1)(n - 2)...(n -r +1) n-r =n.

It will be convenient to remember both these expressions for nCr, using (1) in all cases where a numerical result is required, and (2) when it is sufficient to leave it in an algebraic shape.