A college algebra

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ...may be written--n n--1 n--2 n--r + 1 C.,r-J x--2 x--3--x. x The numerators of the factors on the right side of this equation begin with n, and form a descending series with the common difference 1; and the denominators begin with 1, and form an ascending series with the common difference 1. Therefore, from some point in the series, these factors become less than 1. Hence, the maximum product is reached when that product includes all the factors greater than 1. 1. When n is an odd number the numerator and the denominator of each factor are alternately both odd and both even, so that the factor greater than 1, but nearest to 1, is the factor whose numerator exceeds the denominator by 2. Hence, in this case, r must have such a value that n--1 n--r + 1 = r + 2, or r =--2. When n is an even number the numerator of the first factor is even and the denominator odd; the numerator of the second factor is odd and the denominator even; and so on, alternately, so that the factor greater than 1, but nearest to 1, is the factor whose numerator exceeds the denominator by 1. Hence, in this case, r must have such a value that n--? + 1 = r + 1, or r =. (1) What value of r will give the greatest number of combinations out of 7 things? Here n is odd, and r =--=--3. 2 2..Ct,1 = 7x6x5 = 35. '1x2x3 Tt A »v o 7 x 6 x 5 x 4 or If r = 4, then C7 4 = = 35. 1x2x3x4 When the number of things is odd there are two equal numbers of combinations, namely, when the number of things taken together is just under axidjust over one-half of the whole number of things. (2) What value of r will give the greatest number of selections out of 8 things? Here n is even, and r =-=-= 4. 2 2 8x7x6x5 '1x2x3x4 so that, when the number of things is even, the number of selections will be greatest when one...