CommunicatingMathematicsThe general expansion of (xy)ncan also be determined by theBinomialTheorem.Use the Binomial Theorem to expand (2xy)6.The expansion will have seven terms. Find the first four terms using thesequence 1,61or 6,6152or 15,615243or 20. Then use symmetry to find theremaining terms, 15, 6, and 1.(2xy)6(2x)66(2x)5(y)15(2x)4(y)220(2x)3(y)315(2x)2(y)46(2x)(y)5(y)664x6192x5y240x4y2160x3y360x2y412xy5y6An equivalent form of the Binomial Theorem uses both sigma and factorialnotation. It is written as follows, wherenis a positive integer andris a positiveinteger or zero.(xy)nnr0r!(nn!r)!xnryrYou can use this form of the Binomial Theorem to find individual terms of anexpansion.Find the fifth term of (4a3b)7.(4a3b)77r0r!(77!r)!(4a)7r(3b)rTo find the fifth term, evaluate the general term forr4.Since r increases from0 to n, r is one less than the number of the term.r!(77!r)!(4a)7r(3b)r4!(77!4)!(4a)74(3b)4764! 35!4!(4a)3(3b)4181,440a3b4The fifth term of (4a3b)7is 181,440a3b4.Lesson 12-6The Binomial Theorem803Ifnis a positive integer, then the following is true.