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. 1907 Excerpt:... one factor and x from the other three; four such terms occur, so the group is ix3. Terms containing x twice occur by taking 1 from any two of the factors and x from the remaining factors; the number of such terms is the number of ways of choosing 2 factors from the 4 factors, that is 4 x 3.r 2, or 6; so the group is 6x. Terms containing x once occur by taking x from any one factor and 1 from the remainder; this one factor can be chosen in 4 ways, so the group is ix. Verify these results by substituting 1 for a b c d in the expression found for (x + a) (x + b) (x + c) (x + d), and also by simply multiplying out (x+1)4. 20. Give the various groups of terms when x + a, x+b, x + c, x + d, x + c, x+f, x+g are multiplied together. How many terms are there in each group? Also find the number of terms by multiplying out (x+lf, and check the result by actually multiplying together x + a, x + b, &c. 21. When n factors x + a, x + b,... x + t are multiplied together, how many terms are there in the groups containing x1, a?1-1, xn2, x"3, a?1-4? The numbers of terms are n(n-l) w(n-l)(n-2) w(w-l) (n-2) (w-3) 'n, 2' 2x3' 2x3x4 When (x+l)n is multiplied out, write down the first five groups. They are r» N(n-i) 2 n(n-l)(«-2) 3 x' " '--2 ' 2x3' n(n-!) (w-2)(n-3) 4 2x3x4 The result that (x+l)n=xn+nxn-1+x+... The remaining articles of this chapter should be postponed till they are wanted in article 4 of chapter XII. is known as the binomial theorem, and the series xn + nxn1 +... is called the binomial series. 22. When the n factors x + a,... x + t are multiplied together, how many terms are there that do not contain x, that contain x once, twice, &c.? When (x + l)n is multiplied out, what are the last few groups? They are n(n-l)(n-2) n(n-l) 2x3 ' 2...