Ifn is a positive integer and we multiply out (r+y) term by term, each term will...

 

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Ifn is a positive integer and we multiply out (r+y)" term by term, each term will be the product of x's and y's, with an x or a y coming from each of then factors x+y. For instance, the expansion (x+y) = (x+y)(x+y)(r +y) %3! =x.X.x+xx y+x.y.x+x•y.y +y x.x+y x•y+y y x+y y.y ='+3r*y+3ry +y yields terms of the form r.ry, xy, and y'. Their coefficients are 1, 3, 3, and 1, and the coefficient of xy, for example, is G) = 3, the number of ways in which we can choose the two factors providing the y's. Similarly, the coefficient of ry is G) = 3. the number of ways in which we can choose the one factor providing the y, and the coefficients of r and y are (2) =1 and () = 1. More generally, if n is a positive integer and we multiply out (x+y)" term by term, the coefficient of "-y is ("), the number of ways in which we can choose the r factors providing the y's. Accordingly, we refer to Values of the binomial coefficients for n = 0,1,...,20 and r = 0,1,..., 10 are given in table Factorials and Binomial Coefficients of "Statistical Tables." We can now as a binomial coefficient. state the following theorem. THEOREM 9. (x+y)" = py for any positive integer n 17. Show that =n2-1 by setting x = 1 in The- orem 9, then differentiating the expressions on both sides with respect to y, and finally substituting y= 1. Ifn is a positive integer and we multiply out (r+y)" term by term, each term will be the product of x's and y's, with an x or a y coming from each of then factors x+y. For instance, the expansion (x+y) = (x+y)(x+y)(r +y) %3! =x.X.x+xx y+x.y.x+x•y.y +y x.x+y x•y+y y x+y y.y ='+3r*y+3ry +y yields terms of the form r.ry, xy, and y'. Their coefficients are 1, 3, 3, and 1, and the coefficient of xy, for example, is G) = 3, the number of ways in which we can choose the two factors providing the y's. Similarly, the coefficient of ry is G) = 3. the number of ways in which we can choose the one factor providing the y, and the coefficients of r and y are (2) =1 and () = 1. More generally, if n is a positive integer and we multiply out (x+y)" term by term, the coefficient of "-y is ("), the number of ways in which we can choose the r factors providing the y's. Accordingly, we refer to Values of the binomial coefficients for n = 0,1,...,20 and r = 0,1,..., 10 are given in table Factorials and Binomial Coefficients of "Statistical Tables." We can now as a binomial coefficient. state the following theorem. THEOREM 9. (x+y)" = py for any positive integer n 17. Show that =n2-1 by setting x = 1 in The- orem 9, then differentiating the expressions on both sides with respect to y, and finally substituting y= 1.