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

6 1
6
Example 1
b. (3 x 2 y )
Extend Pascal’s triangle to the eighth row.
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Then, write the expression and simplify each term. Replace each x with 3
and y with 2 y
7
x
.
4
4
FAMILY
Refer to the application at the beginning of the lesson. Of the seven
children born to the McCaughey’s, at least three were boys. How many of
the possible groups of boys and girls have at least three boys?
Example 2
Let g represent girls and b represent boys. To find the number of possible
groups, expand ( g b )
7
. Use the eighth row of Pascal’s triangle for the
expansion.
7
1 or 99.
802
Chapter 12 Sequences and Series

Communicating
Mathematics
The general expansion of ( x y )
n
can also be determined by the Binomial
Theorem.
Use the Binomial Theorem to expand (2 x y )
6
.
The expansion will have seven terms. Find the first four terms using the
sequence 1,
6
1
or 6,
6
1
5
2
or 15,
6
1
5
2
4
3
or 20. Then use symmetry to find the
remaining terms, 15, 6, and 1.
(2 x y )
6
(2 x )
6
6(2 x )
5
( y ) 15(2 x )
4
( y )
2
20(2 x )
3
( y )
3
15(2 x )
2
( y )
4
6(2 x )( y )
5
( y )
6
64 x
6
192 x
5
y 240 x
4
y
2
160 x
3
y
3
60 x
2
y
4
12 xy
5
y
6
An equivalent form of the Binomial Theorem uses both sigma and factorial
notation. It is written as follows, where n is a positive integer and r is a positive
integer or zero.
( x y )
n
n
r 0
r !( n
n !
r )!
x
n r
y
r
You can use this form of the Binomial Theorem to find individual terms of an
expansion.
Find the fifth term of (4 a 3 b )
7
.
(4 a 3 b )
7
7
r 0
r !(7
7!
r )!
(4 a )
7 r
(3 b )
r
To find the fifth term, evaluate the general term for r 4. Since r increases from
0 to n, r is one less than the number of the term.
r !(7
7!
r )!
(4 a )
7 r
(3 b )
r
4!(7
7!
4)!
(4 a )
7 4
(3 b )
4
7 6
4! 3
5
!
4!
(4 a ) 3 (3 b ) 4
181,440 a
3
b
4
The fifth term of (4 a 3 b )
7
is 181,440 a
3
b
4
.
Lesson 12-6 The Binomial Theorem
803
If n is a positive integer, then the following is true.