14. Substituting appropriate values for x and y into the formula of Theorem 9, show that...

 

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14. Substituting appropriate values for x and y into the formula of Theorem 9, show that n (a) > 2": r=0 (b) -1 = 0; r=0 (c) (a r=0 the number of ways in which we can choose the one factor providing the y, and the ) = 1 and () = 1. %3D coefficients of x and y are ( More generally, if n is a positive integer and we multiply out (x+y)" term by term, the coefficient of x"-ry is ("), the number of ways in which we can choose (の) as a binomial coefficient. the r factors providing the y's. Accordingly, we refer to 0,1,...,20 and r 0,1,..., 10 are given Values of the binomial coefficients for n = in table Factorials and Binomial Coefficients of "Statistical Tables." We can now state the following theorem. 39 THEOREM 9. (x+y)" = } x"y for any positive integer n 11 Introduction DEFINITION 3. BINOMIAL COEFFICIENTS. The coefficient of x"-ry in the binomial expansion of (x+y)" is called the binomial coefficient (). The calculation of binomial coefficients can often be simplified by making use of the three theorems that follow. THEOREM 10. For any positive integers n and r= 0, 1, 2, .. ,n, 99+ 78°F DI 14. Substituting appropriate values for x and y into the formula of Theorem 9, show that n (a) > 2": r=0 (b) -1 = 0; r=0 (c) (a r=0 the number of ways in which we can choose the one factor providing the y, and the ) = 1 and () = 1. %3D coefficients of x and y are ( More generally, if n is a positive integer and we multiply out (x+y)" term by term, the coefficient of x"-ry is ("), the number of ways in which we can choose (の) as a binomial coefficient. the r factors providing the y's. Accordingly, we refer to 0,1,...,20 and r 0,1,..., 10 are given Values of the binomial coefficients for n = in table Factorials and Binomial Coefficients of "Statistical Tables." We can now state the following theorem. 39 THEOREM 9. (x+y)" = } x"y for any positive integer n 11 Introduction DEFINITION 3. BINOMIAL COEFFICIENTS. The coefficient of x"-ry in the binomial expansion of (x+y)" is called the binomial coefficient (). The calculation of binomial coefficients can often be simplified by making use of the three theorems that follow. THEOREM 10. For any positive integers n and r= 0, 1, 2, .. ,n, 99+ 78°F DI