# 1 6 15 20 15 6 1 x y 6 x 6 6 x 5 y 15 x 4 y 2 20 x 3

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616Example1
b. (3x2y)Extend Pascal’s triangle to the eighth row.1615201561172135352171Then, write the expression and simplify each term. Replace eachxwith 3andywith 2y7x.44
FAMILYRefer to the application at the beginning of the lesson. Of the sevenchildren born to the McCaughey’s, at least three were boys. How many ofthe possible groups of boys and girls have at least three boys?Example2
Letgrepresent girls andbrepresent boys. To find the number of possiblegroups, expand (gb)7. Use the eighth row of Pascal’s triangle for theexpansion.71 or 99.
802Chapter 12Sequences and Series
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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.

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Chapter 1 / Exercise 1
Elementary Algebra
Tussy/Gustafson
Expert Verified