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1012 Chapter 11 Sequences, Induction, and Probability
Section
11.5 The Binomial Theorem
Objectives
Galaxies are groupings of billions of stars
bound together by gravity. Some galaxies,
such as the Centaurus galaxy shown
here, are elliptical in shape.
� Evaluate a binomial
�
�
coefficient.
Expand a binomial raised to a
power.
Find a particular term in a
binomial expansion.
I
s mathematics discovered or
invented? For example, planets
revolve in elliptical orbits. Does that
mean that the ellipse is out there,
waiting for the mind to discover it? Or
do people create the definition of an
ellipse just as they compose a song? And is it
possible for the same mathematics to be
discovered/invented by independent researchers
separated by time, place, and culture? This is precisely what occurred when
mathematicians attempted to find efficient methods for raising binomials to higher
and higher powers, such as
1x + 223, 1x + 224, 1x + 225, 1x + 226,
and so on. In this section, we study higher powers of binomials and a method first
discovered/invented by great minds in Eastern and Western cultures working
independently.
�
Evaluate a binomial coefficient.
Binomial Coefficients
Before turning to powers of binomials, we introduce a special notation that uses
factorials.
n
r
Definition of a Binomial Coefficient ¢ ≤
n
For nonnegative integers n and r, with n Ú r, the expression ¢ ≤ (read “n
r
above r”) is called a binomial coefficient and is defined by
n
n!
a b =
.
r
r!1n - r2!
Technology
Graphing utilities can compute binomial coefficients. For example, to
6
find ¢ ≤ , many utilities require the
2
sequence
6 冷nC r 冷 2 冷ENTER 冷.
The graphing utility will display 15.
Consult your manual and verify the
other evaluations in Example 1.
n
The symbol nCr is often used in place of ¢ ≤ to denote binomial coefficients.
r
EXAMPLE 1
Evaluate:
Evaluating Binomial Coefficients
6
a. ¢ ≤
2
3
b. ¢ ≤
0
9
c. ¢ ≤
3
4
d. ¢ ≤ .
4
Solution In each case, we apply the definition of the binomial coefficient.
6!
6 # 5 # 4!
6
6!
= 15
=
= # #
a. a b =
2!16 - 22!
2!4!
2 1 4!
2
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Section 11.5 The Binomial Theorem
1013
3!
1
3!
3
b. ¢ ≤ = 0!(3-0)! = 0!3! = 1 =1
0
Remember that 0! = 1.
9
9!
9!
9 # 8 # 7 # 6!
c. a b =
=
= # # #
= 84
3
3!19 - 32!
3!6!
3 2 1 6!
4
4!
4!
1
d. a b =
=
= = 1
4
4!14 - 42!
4! 0!
1
Check Point
6
a. ¢ ≤
3
�
Expand a binomial raised
to a power.
1
Evaluate:
6
b. ¢ ≤
0
8
c. ¢ ≤
2
3
d. ¢ ≤ .
3
The Binomial Theorem
When we write out the binomial expression 1a + b2n, where n is a positive integer, a
number of patterns begin to appear.
1a + b21 = a + b
1a + b22 = a2 + 2ab + b2
1a + b23 = a3 + 3a2 b + 3ab2 + b3
1a + b24 = a4 + 4a3 b + 6a2 b2 + 4ab3 + b4
1a + b25 = a5 + 5a4 b + 10a3 b2 + 10a2 b3 + 5ab4 + b5
Each expanded form of the binomial expression is a polynomial. Observe the
following patterns:
1. The first term in the expansion of 1a + b2n is a n. The exponents on a decrease
by 1 in each successive term.
2. The exponents on b in the expansion of 1a + b2n increase by 1 in each successive
term. In the first term, the exponent on b is 0. (Because b0 = 1, b is not shown
in the first term.) The last term is bn.
3. The sum of the exponents on the variables in any term in the expansion of
1a + b2n is equal to n.
4. The number of terms in the polynomial expansion is one greater than the
power of the binomial, n. There are n + 1 terms in the expanded form of
1a + b2n.
Using these observations, the variable parts of the expansion of 1a + b26 are
a6,
a5b,
a4 b2,
a3 b3,
a2 b4,
ab5,
b6.
The first term is a 6, with the exponents on a decreasing by 1 in each successive term.
The exponents on b increase from 0 to 6, with the last term being b6. The sum of the
exponents in each term is equal to 6.
We can generalize from these observations to obtain the variable parts of the
expansion of 1a + b2n. They are
an,
an –1b,
Sum of exponents:
n−1+1=n
an –2b2,
an –3b3, . . . ,
Sum of exponents:
n−3+3=n
abn –1,
bn.
Sum of exponents:
1+n−1=n
Exponents on a are
decreasing by 1.
Exponents on b are
increasing by 1.
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1014 Chapter 11 Sequences, Induction, and Probability
If we use binomial coefficients and the pattern for the variable part of each
term, a formula called the Binomial Theorem can be used to expand any positive
integral power of a binomial.
A Formula for Expanding Binomials: The Binomial Theorem
For any positive integer n,
n
n
n
n
n
1a + b2n = ¢ ≤ an + ¢ ≤ an - 1 b + ¢ ≤ an - 2 b2 + ¢ ≤ an - 3b3 + Á + ¢ ≤ bn
0
1
2
3
n
n
n
= a ¢ ≤ an - r br.
r=0 r
Using the Binomial Theorem
EXAMPLE 2
Expand:
1x + 224.
Solution We use the Binomial Theorem
n
n
n
n
n
1a + b2n = ¢ ≤ an + ¢ ≤ an - 1 b + ¢ ≤ an - 2b2 + ¢ ≤ an - 3 b3 + Á + ¢ ≤ bn
0
1
2
3
n
to expand 1x + 224. In 1x + 224, a = x, b = 2, and n = 4. In the expansion, powers
of x are in descending order, starting with x4. Powers of 2 are in ascending order,
starting with 2 0. (Because 2 0 = 1, a 2 is not shown in the first term.) The sum of the
exponents on x and 2 in each term is equal to 4, the exponent in the expression
1x + 224.
Technology
4
4
4
4
4
(x+2)4=a bx4+a bx3 ⴢ 2+a bx2 ⴢ 22+a bx ⴢ 23+a b24
0
1
2
3
4
You can use a graphing utility’s table
feature to find the five binomial
coefficients in Example 2.
These binomial coefficients are evaluated using A nr B =
Enter y1 = 4 nCr x.
=
A 04 B
A 42 B
A 44 B
A 41 B
n!
r!(n − r)! .
4! 3
4! 2
4!
4!
4! 4
x+
x ⴢ 2+
x ⴢ 4+
x ⴢ 8+
ⴢ 16
1!3!
2!2!
3!1!
4!0!
0!4!
4!
2!2!
=
4 ⴢ 3 ⴢ 2!
2! ⴢ 2 ⴢ 1
=
12
2
=6
Take a few minutes to verify the other factorial evaluations.
A 43 B
=1 ⴢ x4+4x3 ⴢ 2+6x2 ⴢ 4+4x ⴢ 8+1 ⴢ 16
= x4 + 8x3 + 24x2 + 32x + 16
Check Point
EXAMPLE 3
Expand:
2
Expand:
1x + 124.
Using the Binomial Theorem
12x - y25.
Solution Because the Binomial Theorem involves the addition of two terms
raised to a power, we rewrite 12x - y25 as 32x + 1-y245. We use the Binomial
Theorem
n
n
n
n
n
1a + b2n = ¢ ≤ an + ¢ ≤ an - 1 b + ¢ ≤ an - 2b2 + ¢ ≤ an - 3 b3 + Á + ¢ ≤ bn
0
1
2
3
n
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Section 11.5 The Binomial Theorem
1015
to expand 32x + 1-y245. In 32x + 1- y245, a = 2x, b = - y, and n = 5. In the
expansion, powers of 2x are in descending order, starting with 12x25. Powers of -y
are in ascending order, starting with 1-y20. [Because 1-y20 = 1, a - y is not shown
in the first term.] The sum of the exponents on 2x and - y in each term is equal to 5,
the exponent in the expression 12x - y25.
12x - y25 = 32x + 1- y245
5
5
5
5
5
5
=a b(2x)5+a b(2x)4(–y)+a b(2x)3(–y)2+a b(2x)2(–y)3+a b(2x)(–y)4+a b(–y)5
0
1
2
3
4
5
Evaluate binomial coefficients using A nr B =
=
n!
r!(n − r)! .
5!
5!
5!
5!
5!
5!
(2x)5+
(2x)4(–y)+
(2x)3(–y)2+
(2x)2(–y)3+
(2x)(–y)4+
(–y)5
1!4!
0!5!
2!3!
3!2!
4!1!
5!0!
5!
2!3!
=
5 ⴢ 4 ⴢ 3!
2 ⴢ 1 ⴢ 3!
= 10
Take a few minutes to verify the other factorial evaluations.
=1(2x)5+5(2x)4(–y)+10(2x)3(–y)2+10(2x)2(–y)3+5(2x)(–y)4+1(–y)5
Raise both factors in these parentheses to the indicated powers.
=1(32x5)+5(16x4)(–y)+10(8x3)(–y)2+10(4x2)(–y)3+5(2x)(–y)4+1(–y)5
Now raise −y to the indicated powers.
= 1132x52 + 5116x421 - y2 + 1018x32y2 + 1014x221 - y32 + 512x2y4 + 11 -y52
Multiplying factors in each of the six terms gives us the desired expansion:
12x - y25 = 32x5 - 80x4 y + 80x3 y2 - 40x2 y3 + 10xy4 - y5.
Check Point
�
Find a particular term in a
binomial expansion.
3
Expand:
1x - 2y25.
Finding a Particular Term in a Binomial Expansion
By observing the terms in the formula for expanding binomials, we can find a formula
for finding a particular term without writing the entire expansion.
1st term
n
a banb0
0
2nd term
3rd term
n
a ba n –1b1
1
n
a ba n–2b2
2
The exponent on b is 1 less than the term number.
Based on the observation in the bottom voice balloon, the 1r + 12st term of the
expansion of 1a + b2n is the term that contains br.
Finding a Particular Term in a Binomial Expansion
The 1r + 12st term of the expansion of 1a + b2n is
n
r
¢ ≤ an - rbr.
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1016 Chapter 11 Sequences, Induction, and Probability
Finding a Single Term of a Binomial Expansion
EXAMPLE 4
Find the fourth term in the expansion of 13x + 2y27.
Solution The fourth term in the expansion of 13x + 2y27 contains 12y23. To find
the fourth term, first note that 4 = 3 + 1. Equivalently, the fourth term of
13x + 2y27 is the 13 + 12st term. Thus, r = 3, a = 3x, b = 2y, and n = 7. The
fourth term is
7!
7
7
a b(3x)7–3(2y)3=a b(3x)4(2y)3=
(3x)4(2y)3.
3
3
3!(7-3)!
We use A nr B =
Use the formula for
the (r + 1)st term of
(a + b)n:
A nr B an− rbr.
to
n!
r!(n − r)!
evaluate A 37 B.
Now we need to evaluate the factorial expression and raise 3x and 2y to the indicated
powers. We obtain
7!
7 # 6 # 5 # 4!
181x4218y32 = # # #
181x4218y32 = 35181x4218y32 = 22,680x4y3.
3!4!
3 2 1 4!
The fourth term of 13x + 2y27 is 22,680x4 y3.
Check Point
4
Find the fifth term in the expansion of 12x + y29.
The Universality of Mathematics
Pascal’s triangle is an array of numbers showing coefficients of the terms in the expansions of 1a + b2n. Although credited to
French mathematician Blaise Pascal (1623–1662), the triangular array of numbers appeared in a Chinese document printed
in 1303. The Binomial Theorem was known in Eastern cultures prior to its discovery in Europe. The same mathematics is often
discovered/invented by independent researchers separated by time, place, and culture.
Binomial Expansions
Pascal’s Triangle
1a + b20 = 1
1
1a + b2 = a + b
1
1
1a + b22 = a2 + 2ab + b2
1
1a + b2 = a + 3a b + 3ab + b
3
3
2
2
3
1
1a + b24 = a4 + 4a3b + 6a2 b2 + 4ab3 + b4
1
1a + b2 = a + 5a b + 10a b + 10a b + 5ab + b
5
Chinese Document: 1303
Coefficients in the Expansions
5
4
3 2
2 3
4
5
1
1
1
1 8
6
7
2
3
4
5
1
1
3
6
10 10
15 20
1
4
1
5
1
15 6
1
21 35 35 21 7
28 56 70 56 28 8
1
1
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Section 11.5 The Binomial Theorem
1017
Exercise Set 11.5
Practice Exercises
1
In Exercises 1–8, evaluate the given binomial coefficient.
8
1. ¢ ≤
3
4. ¢
11
≤
1
100
7. ¢
≤
2
7
2. ¢ ≤
2
3. ¢
6
5. ¢ ≤
6
6. ¢
12
≤
1
15
≤
2
100
8. ¢
≤
98
In Exercises 9–30, use the Binomial Theorem to expand each
binomial and express the result in simplified form.
9. 1x + 223
10. 1x + 423
13. 15x - 123
14. 14x - 123
11. 13x + y23
15. 12x + 124
19. 1y - 324
4
21. 12x3 - 12
18. 1x2 + y2
4
22. 12x5 - 12
4
23. 1c + 22
4
26. 1x - 225
27. 13x - y2
5
29. 12a + b2
30. 1a + 2b26
6
32. 1x + 328
33. 1x - 2y210
38. 1y3 - 12
20
In Exercises 39–48, find the term indicated in each expansion.
39. 12x + y2 ; third term
40. 1x + 2y2 ; third term
43. 1x + y 2 ; sixth term
44. 1x + y 2 ; sixth term
41. 1x - 129; fifth term
2
45. A x -
x 10
3
+ b .
x
3
56. Find the middle term in the expansion of a
12
1
- x2 b .
x
3 8
B fourth term
1 9
2 ;
6
42. 1x - 1210; fifth term
3
46. A x +
47. 1x2 + y2 ; the term containing y 14
22
2 8
B fourth term
1 8
2 ;
48. 1x + 2y210; the term containing y6
0.28
0.12
0.20
0.10
0.19
Anxiety/Panic
Disorder
0.08
0.14
0.07
0.05
0.15
0.20
Probability
0.25
0.30
If the probability an event will occur is p and the probability it
will not occur is q, then each term in the expansion of 1p + q2n
represents a probability.
57. The probability that a smoker suffers from depression is 0.28.
If five smokers are randomly selected, the probability that
three of them will suffer from depression is the third term of
the binomial expansion of
(0.28 + 0.72) .
In Exercises 49–52, use the Binomial Theorem to expand each
expression and write the result in simplified form.
4
0.10
5
Practice Plus
49. 1x3 + x -22
General Population
Source: MARS 2005 OTC/DTC
21
6
and simplify.
54. f1x2 = x 5 + 8
17
37. 1y3 - 12
1 3
≤
1
3x
55. Find the middle term in the expansion of a
0
36. 1x2 + 12
16
53. f1x2 = x 4 + 7
Severe
Pain
34. 1x - 2y29
35. 1x2 + 12
h
Frequent
Hangovers
In Exercises 31–38, write the first three terms in each binomial
expansion, expressing the result in simplified form.
31. 1x + 228
f1x + h2 - f1x2
Depression
28. 1x - 3y2
5
In Exercises 53–54, find
Tobacco-Dependent Population
5
25. 1x - 125
2
52. ¢ x3 -
Probability That United States Adults Suffer from Various Ailments
24. 1c + 32
5
B
1 3
3
The graph shows that U.S. smokers have a greater probability of
suffering from some ailments than the general adult population.
Exercises 57–58 are based on some of the probabilities, expressed
as decimals, shown to the right of the bars. In each exercise, use a
calculator to determine the probability, correct to four decimal
places.
16. 13x + 124
20. 1y - 424
-
Application Exercises
12. 1x + 3y23
17. 1x2 + 2y2
51. A x3 - x
50. 1x2 + x -32
4
Probability a
smoker suffers
from depression
What is this probability?
5 smokers
are selected.
Probability a
smoker does not
suffer from depression
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1018 Chapter 11 Sequences, Induction, and Probability
58. The probability that a person in the general population suffers from depression is 0.12. If five people from the general
population are randomly selected, the probability that three
of them will suffer from depression is the third term of the
binomial expansion of
5
(0.12 + 0.88) .
Probability a person in
the general population
suffers from depression
5 people from the general
population are selected.
Probability a person in
the general population does
not suffer from depression
69. f11x2 = 1x + 124
f31x2 = x + 4x
4
f21x2 = x4
f41x2 = x4 + 4x3 + 6x2
3
f51x2 = x4 + 4x3 + 6x2 + 4x
f61x2 = x4 + 4x3 + 6x2 + 4x + 1
Use a 3-5, 5, 14 by 3-30, 30, 104 viewing rectangle.
In Exercises 70–72, use the Binomial Theorem to find a
polynomial expansion for each function. Then use a graphing
utility and an approach similar to the one in Exercises 68 and 69
to verify the expansion.
70. f11x2 = 1x - 123
71. f11x2 = 1x - 224
72. f11x2 = 1x + 226
What is this probability?
Writing in Mathematics
Critical Thinking Exercises
n
59. Explain how to evaluate ¢ ≤ . Provide an example with your
r
explanation.
60. Describe the pattern on the exponents on a in the expansion
of 1a + b2n.
Make Sense? In Exercises 73–76, determine whether each
statement makes sense or does not make sense, and explain
your reasoning.
62. What is true about the sum of the exponents on a and b in
any term in the expansion of 1a + b2n?
n
75. I use binomial coefficients to expand 1a + b2n, where ¢ ≤ is
1
n
the coefficient of the first term, ¢ ≤ is the coefficient of the
2
second term, and so on.
7
76. One of the terms in my binomial expansion is ¢ ≤ x2y4.
5
61. Describe the pattern on the exponents on b in the expansion
of 1a + b2n.
63. How do you determine how many terms there are in a
binomial expansion?
64. Explain how to use the Binomial Theorem to expand a
binomial. Provide an example with your explanation.
65. Explain how to find a particular term in a binomial expansion
without having to write out the entire expansion.
66. Describe how you would use mathematical induction to prove
n
n
n
1a + b2n = ¢ ≤ an + ¢ ≤ an - 1 b + ¢ ≤ an - 2 b2
0
1
2
+ Á + ¢
5
74. Without writing the expansion of 1x - 126, I can see that the
terms have alternating positive and negative signs.
In Exercises 77–80, determine whether each statement is true or
false. If the statement is false, make the necessary change(s) to
produce a true statement.
77. The binomial expansion for 1a + b2n contains n terms.
78. The Binomial Theorem can be written in condensed form as
n
n
≤ abn - 1 + ¢ ≤ bn.
n - 1
n
What happens when n = 1? Write the statement that we
assume to be true. Write the statement that we must prove.
What must be done to the left side of the assumed statement
to make it look like the left side of the statement that must
be proved? (More detail on the actual proof is found in
Exercise 85.)
67. Use the 冷nCr 冷 key on a graphing utility to verify your answers
in Exercises 1–8.
In Exercises 68–69, graph each of the functions in the same
viewing rectangle. Describe how the graphs illustrate the Binomial
Theorem.
68. f11x2 = 1x + 223
f31x2 = x3 + 6x2
n
n
1a + b2n = a ¢ ≤ an - r br.
r=0 r
79. The sum of the binomial coefficients in 1a + b2n cannot be 2 n.
80. There are no values of a and b such that
1a + b24 = a4 + b4.
81. Use the Binomial Theorem to expand and then simplify the
3
result: 1x2 + x + 12 .
Hint: Write x2 + x + 1 as x2 + 1x + 12.
Technology Exercises
82. Find the term in the expansion of 1x2 + y22 containing x4 as
a factor.
f41x2 = x3 + 6x2 + 12x
2
Use a 3 -10, 10, 14 by 3 - 30, 30, 104 viewing rectangle.
5
83. Prove that
n
r
¢ ≤ = ¢
n
≤.
n - r
84. Show that
n
r
f21x2 = x3
f51x2 = x + 6x + 12x + 8
3
73. In order to expand 1x3 - y42 , I find it helpful to rewrite the
expression inside the parentheses as x3 + 1 -y42.
¢ ≤ + ¢
Hints:
n
n + 1
≤ = ¢
≤.
r + 1
r + 1
1n - r2! = 1n - r21n - r - 12!
1r + 12! = 1r + 12r!
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Section 11.6 Counting Principles, Permutations, and Combinations
85. Follow the outline below and use mathematical induction to
prove the Binomial Theorem:
k
k + 1
k
k + 1
f. Because ¢ ≤ = ¢
≤ (why?) and ¢ ≤ = ¢
≤
0
0
k
k + 1
n
n
n
1a + b2n = ¢ ≤ an + ¢ ≤ an - 1 b + ¢ ≤ an - 2b2
0
1
2
+ Á + ¢
1019
(why?), substitute these results and the results from part
(e) into the equation in part (d). This should give the
statement that we were required to prove in the second
step of the mathematical induction process.
n
n
≤ abn - 1 + ¢ ≤ bn.
n - 1
n
a. Verify the formula for n = 1.
b. Replace n with k and write the statement that is assumed
true. Replace n with k + 1 and write the statement that
must be proved.
Preview Exercises
Exercises 86–88 will help you prepare for the material covered in
the next section.
c. Multiply both sides of the statement assumed to be true
by a + b. Add exponents on the left. On the right,
distribute a and b, respectively.
86. Evaluate
n!
for n = 20 and r = 3.
1n - r2!
d. Collect like terms on the right. At this point, you
should have
87. Evaluate
n!
for n = 8 and r = 3.
1n - r2! r!
k
k
k
1a + b2k + 1 = ¢ ≤ ak + 1 + B ¢ ≤ + ¢ ≤ R akb
0
0
1
k
k
k
k
+ B ¢ ≤ + ¢ ≤ R a k - 1 b2 + B ¢ ≤ + ¢ ≤ R a k - 2 b3
1
2
2
3
+ Á + B¢
88. You can choose from two pairs of jeans (one blue, one black)
and three T-shirts (one beige, one yellow, and one blue), as
shown in the diagram.
k
k
k
≤ + ¢ ≤ R abk + ¢ ≤ bk + 1.
k - 1
k
k
e. Use the result of Exercise 84 to add the binomial sums in
n
n
brackets. For example, because ¢ ≤ + ¢
≤
r
r + 1
= ¢
n + 1
k
k
k + 1
≤ , then ¢ ≤ + ¢ ≤ = ¢
≤ and
r + 1
0
1
1
k
1
k
2
¢ ≤ + ¢ ≤ = ¢
Section
k + 1
≤.
2
11.6 Counting Principles, Permutations, and Combinations
Objectives
� Use the Fundamental
�
�
�
True or false: The diagram shows that you can form 2 * 3, or
6, different outfits.
Counting Principle.
Use the permutations
formula.
Distinguish between
permutation problems and
combination problems.
Use the combinations
formula.
H
ave you ever imagined what your life would be like if you
won the lottery? What changes would you make? Before
you fantasize about becoming a person of leisure with a
staff of obedient elves, think about this: The probability
of winning top prize in the lottery is about the
same as the probability of being struck by
lightning. There are millions of possible
number combinations in lottery games
and only one way of winning the grand
prize. Determining the probability of winning involves calculating the chance of
getting the winning combination from all
possible outcomes. In this section, we begin
preparing for the surprising world of probability by
looking at methods for counting possible outcomes.