   Chapter 14.1, Problem 94E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# DISCUSS ■ DISCOVER: Why Is ( n r ) the Same as C ( n , r ) ? This exercise explains why the binomial coefficients ( n r ) that appear in the expression of ( x + y ) n are the same as C ( n , r ) , the number of ways of choosing r objects from n objects. First, note that expanding a binomial using only the Distributive Property gives ( x + y ) 2 = ( x + y ) ( x + y ) = ( x + y ) x + ( x + y ) y = x x + x y + y x + y y ( x + y ) 3 = ( x + y ) ( x x + x y + y x + y y ) = x x x + x x y + x y x + x y y + y x x + y x y + y y x + y y y (a) Expand ( x + y ) 5 using only the Distribution Property.(b) Write all the terms that represent x 2 y 3 . These are all the terms that contain two x ’s and three y ’s.(c) Note that the two x ’s appear in all possible positions. Conclude that the number of terms that represent x 2 y 3 is C ( 5 , 2 ) .(d) In general, explain why ( n r ) in the Binomial Theorem is the same as C ( n , r ) .

To determine

(a)

To find:

The expansion of the expression using only the Distributive Property.

Explanation

Given:

The expression is (x+y)5 and expansion by using the Distributive property is,

(x+y)2=(x+y)(x+y)=(x+y)x+(x+y)y=xx+xy+yx+yy(x+y)3=(x+y)(xx+xy+yx+yy)=xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy

Calculation:

Expand the expression (x+y)5 by using the Distributive property.

(x+y)5=(x+y)(x+y)(x+y)(x+y)(x+y)=(x+y)(x+y)(x+y)x+(x+y)(x+y)(x+y)y=[(x+y)(x+y)(x+y)xx+(x+y)(x+y)(x+y)yx+(x+y)(x+y)(x+y)xy+(x+y)(x+y)(x+y)yy]

Use the Distributive Property again.

(x+y)5=[(x+y)(x+y)(x+y)xx+(x+y)(x+y)(x+y)yx+(x+y)(x+y)(x+y)xy+(x+y)(x+y)(x+y)yy]=[(x+y)(x+y)xxx+(x+y)(x+y)yxx+(x+y)(x+y)xyx+(x+y)(x+y)yyx+(x+y)(x+y)xxy+(x+y)(x+y)yxy+(x+y)(x+y)xyy+(x+y)(x+y)yyy]

Again use the Distributive Property

To determine

(b)

To find:

All the terms that represent x2y3 together.

To determine

(c)

To conclude:

The number of terms representing x2y3 is C(5,2).

To determine

(d)

To explain:

The reason that (nr) in the Binomial theorem is same as C(n,r).

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