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PERMUTATIONS AND COMBINATIONS. 329 ally increased, until n+1 r - 1 becomes equal to 1 or less than 1.

Now n+1 r 1 > 1, so long as n+1 r > 2; that is, n+1 2 > r.

We have to choose the greatest value of r consistent with this inequality.

(1) Let n be even, and equal to 2m; then

n+1 2 = 2m+1 2 = m+1 2,

and for all values of r up to m inclusive this is greater than r. Hence by putting r = m = n 2, we find that the greatest number of combinations is nCn 2.

(2) Let n be odd, and equal to 2m+1 ; then

n+1 2 = 2m+2 2 = m+1;

and for all values of r up to m inclusive this is greater than r; but when r=m+1, the multiplying factor becomes equal to 1, and

nCm+1 = nCm; that is, nCn+1 2 = nCn-1 2 ;

and therefore the number of combinations is greatest when the things are taken n+1 2, or n-1 2 at a time; the result being the same in the two cases.

EXAMPLES XXXV. b.

1. Find the number of permutations which can be made from all the letters of the words,

(1) irresistible, (2) phenomenon, (3) tittle-tattle.

2. How many different numbers can be formed by using the seven digits 2, 3, 4, 3, 3, 1, 2 ? How many with the digits 2, 3, 4, 3, 3, 0, 2 ?