1075 11.4 The Binomial Theorem The last term (the sixth term) can be written as y5 = 1x0y5, with coefficient 1 and exponents of 0 and 5. By definition 0! = 1, so 1 = 5! 0! 5! . Generalizing from these examples, the coefficient for the term of the expansion of 1x + y2n in which the variable part is xryn-r (where r … n) is n! r!1n −r2! . This number, called a binomial coefficient, is often symbolized An r B or nCr (read “ n choose r ”). Binomial Coefficient For nonnegative integers n and r, with r … n, the binomial coefficient is defined as follows. nCr = a n rb = n! r!1n −r2! The binomial coefficients are numbers from Pascal’s triangle. For example, A 3 0B is the first number in row three, and A 7 4B is the fifth number in row seven. EXAMPLE 1 Evaluating Binomial Coefficients Evaluate each binomial coefficient. (a) a 6 2b (b) a 8 0b (c) a 10 10b (d) 12C10 ALGEBRAIC SOLUTION (a) a 6 2b = 6! 2!16 - 22! = 6! 2! 4! = 6 # 5 # 4 # 3 # 2 # 1 2 # 1 # 4 # 3 # 2 # 1 = 15 (b) a 8 0b = 8! 0!18 - 02! = 8! 0! 8! = 8! 1 # 8! = 1 0! = 1 (c) a 10 10b = 10! 10!110 - 102! = 10! 10! 0! = 1 0! = 1 (d) 12C10 = 12! 10!112 - 102! = 12! 10! 2! = 66 GRAPHING CALCULATOR SOLUTION Graphing calculators calculate binomial coefficients using the notation nCr . For the TI-84 Plus, this function is found in the MATH menu. Figure 11 shows the values of the binomial coefficients for parts (a)–(d). Compare the results to those in the algebraic solution. Figure 11 S Now Try Exercises 15, 19, and 27.
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