B. Binomial Coefficients and Factorials
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CHAPTER 9 Additional Topics in Algebra
Solution
ᮣ
4 # 3!
4
4!
ϭ
a. a b ϭ
ϭ4
1
1!3!
1!14 Ϫ 12!
4 # 3 # 2!
4
4!
4#3
ϭ
ϭ
ϭ6
b. a b ϭ
2
2!2!
2
2!14 Ϫ 22!
4
4!
4 # 3!
ϭ
ϭ4
c. a b ϭ
3
3!1!
3!14 Ϫ 32!
Now try Exercises 13 through 20
ᮣ
4
4
4
Note a b ϭ 4, a b ϭ 6, and a b ϭ 4 give the interior entries in the fifth row of
1
2
3
Pascal’s triangle: 1 4 6 4 1. For consistency and symmetry, we define 0! ϭ 1, which
enables the formula to generate all entries of the triangle, including the “1’s.”
4!
4
4!
4
4!
ϭ
apply formula a b ϭ
apply formula
a bϭ
4
0
4! # 0!
0!14 Ϫ 02!
4!14 Ϫ 42!
4!
4!
ϭ
0! ϭ 1
ϭ
0! ϭ 1
ϭ1
ϭ1
#
1 4!
4! # 1
n
The formula for a b with 0 Յ r Յ n now gives all coefficients in the 1n ϩ 12st
r
row. For n ϭ 5, we have
5
a b
0
1
EXAMPLE 4
ᮣ
5
a b
1
5
5
a b
2
10
5
a b
3
10
5
a b
4
5
5
a b
5
1
Computing Binomial Coefficients
Compute the binomial coefficients:
9
a. a b
0
Solution
ᮣ
9
b. a b
1
9
9!
a. a b ϭ
0
0!19 Ϫ 02!
9!
ϭ
ϭ1
9!
6
6!
c. a b ϭ
5
5!16 Ϫ 52!
6!
ϭ
ϭ6
5!
6
c. a b
5
6
d. a b
6
9
9!
b. a b ϭ
1
1!19 Ϫ 12!
9!
ϭ
ϭ9
8!
6
6!
d. a b ϭ
6
6!16 Ϫ 62!
6!
ϭ
ϭ1
6!
Now try Exercises 21 through 24
B. You’ve just seen how we
can find binomial coefficients
n
using a b notation
k
ᮣ
n
As mentioned, the formulas for a b and nCr yield like results for given values of n
r
and r. For future use, it will help to commit the general results from Example 4 to
n
n
n
n
memory: a b ϭ 1, a b ϭ n, a
b ϭ n, and a b ϭ 1.
0
1
nϪ1
n
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Section 9.7 The Binomial Theorem
833
C. The Binomial Theorem
n
Using a b notation and the observations made regarding binomial powers, we can now
r
state the binomial theorem.
Binomial Theorem
For any binomial 1a ϩ b2 and natural number n,
n
n
n
1a ϩ b2 n ϭ a b anb0 ϩ a b anϪ1b1 ϩ a b anϪ2b2 ϩ p
0
1
2
n
n
ϩa
b a1bnϪ1 ϩ a b a0bn
nϪ1
n
The theorem can also be stated in summation form as
1a ϩ b2 n ϭ
n
͚ arba
n
nϪr r
b
rϭ0
The expansion actually looks overly impressive in this form, and it helps to summarize the process in words, as we did earlier. The exponents on the first term a begin
at n and decrease, while the exponents on the second term b begin at 0 and increase,
n
keeping the degree of each term constant. The a b notation simply gives the coefficients
r
n
of each term. As a final note, observe that the r in a b gives the exponent on b.
r
EXAMPLE 5
ᮣ
Solution
ᮣ
Expanding a Binomial Using the Binomial Theorem
Expand 1a ϩ b2 6 using the binomial theorem.
6
6
6
6
6
6
6
1a ϩ b2 6 ϭ a b a6b0 ϩ a b a5b1 ϩ a b a4b2 ϩ a b a3b3 ϩ a b a2b4 ϩ a b a1b5 ϩ a b a0b6
0
1
2
3
4
5
6
6! 6
6! 5 1
6! 4 2
6! 3 3
6! 2 4
6! 1 5
6! 6
ϭ
a ϩ
ab ϩ
ab ϩ
ab ϩ
ab ϩ
ab ϩ
b
0!6!
1!5!
2!4!
3!3!
4!2!
5!1!
6!0!
ϭ 1a6 ϩ 6a5b ϩ 15a4b2 ϩ 20a3b3 ϩ 15a2b4 ϩ 6ab5 ϩ 1b6
Now try Exercises 25 through 32
EXAMPLE 6
ᮣ
Using the Binomial Theorem to Find the Initial Terms of an Expansion
Solution
ᮣ
Use the binomial theorem with a ϭ 2x, b ϭ y2, and n ϭ 10.
ᮣ
Find the first three terms of 12x ϩ y2 2 10.
12x ϩ y2 2 10 ϭ a
C. You’ve just seen how
we can use the binomial
theorem to find 1a ؉ b2 n
10
10
10
b12x2 10 1y2 2 0 ϩ a b12x2 9 1y2 2 1 ϩ a b12x2 8 1y2 2 2 ϩ p
0
1
2
10!
ϭ 1121024x10 ϩ 1102512x9y2 ϩ
256x8y4 ϩ p
2!8!
ϭ 1024x10 ϩ 5120x9y2 ϩ 1452256x8y4 ϩ p
ϭ 1024x10 ϩ 5120x9y2 ϩ 11,520x8y4 ϩ p
first three terms
a
10
10
b ϭ 1, a b ϭ 10
0
1
10!
ϭ 45
2!8!
result
Now try Exercises 33 through 36
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CHAPTER 9 Additional Topics in Algebra
D. Finding a Specific Term of the Binomial Expansion
In some applications of the binomial theorem, our main interest is a specific term of the
expansion, rather than the expansion as a whole. To find a specified term, it helps to
consider that the expansion of 1a ϩ b2 n has n ϩ 1 terms: 1a ϩ b2 0 has one term,
n
1a ϩ b2 1 has two terms, 1a ϩ b2 2 has three terms, and so on. Because the notation a b
r
always begins at r ϭ 0 for the first term, the value of r will be 1 less than the term we
are seeking. In other words, for the seventh term of 1a ϩ b2 9, we use r ϭ 6.
The k th Term of a Binomial Expansion
For the binomial expansion 1a ϩ b2 n, the kth term is given by
n
a b anϪrbr, where r ϭ k Ϫ 1.
r
EXAMPLE 7
ᮣ
Solution
ᮣ
Finding a Specific Term of a Binomial Expansion
Find the eighth term in the expansion of 1x ϩ 2y2 12.
By comparing 1x ϩ 2y2 12 to 1a ϩ b2 n we have a ϭ x, b ϭ 2y, and n ϭ 12. Since
we want the eighth term, k ϭ 8 and r ϭ 7. The eighth term of the expansion is
a
D. You’ve just seen how
we can find a specific term
of a binomial expansion
12 5
12!
128x5y7
b x 12y2 7 ϭ
7!5!
7
ϭ 179221128x5y7 2
ϭ 101,376x5y7
27 ϭ 128
1 12
7 2 ϭ 792
result
Now try Exercises 37 through 42
ᮣ
E. Applications
One application of the binomial theorem involves a binomial experiment and binomial
probability. For binomial probabilities, the following must be true: (1) The experiment must have only two possible outcomes, typically called success and failure, and
(2) if the experiment has n trials, the probability of success must be constant for all
n
n trials. If the probability of success for each trial is p, the formula a b11 ؊ p2 n؊kpk
k
gives the probability that exactly k trials will be successful.
Binomial Probability
Given a binomial experiment with n trials, where the probability for success in each
trial is p. The probability that exactly k trials are successful is given by
n
a b11 Ϫ p2 nϪk pk.
k
EXAMPLE 8
ᮣ
Applying the Binomial Theorem — Binomial Probability
Paula Rodrigues has a free-throw shooting average of 85%. On the last play of the
game, with her team behind by three points, she is fouled at the three-point line,
and is awarded two additional free throws via technical fouls on the opposing
coach (a total of five free-throws). What is the probability she makes at least three
(meaning they at least tie the game)?
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Solution
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835
Here we have p ϭ 0.85, 1 Ϫ p ϭ 0.15, and n ϭ 5. The key idea is to recognize the
phrase at least three means “3 or 4 or 5.” So P(at least 3) ϭ P13 ´ 4 ´ 52.
“or” implies a union
P1at least 32 ϭ P13 ´ 4 ´ 52
ϭ P132 ϩ P142 ϩ P152 sum of probabilities (mutually exclusive events)
5
5
5
ϭ a b 10.152 2 10.852 3 ϩ a b 10.152 1 10.852 4 ϩ a b 10.152 0 10.852 5
4
5
3
Ϸ 0.1382 ϩ 0.3915 ϩ 0.4437
ϭ 0.9734
Paula’s team has an excellent chance 1Ϸ97.3% 2 of at least tying the game.
Now try Exercises 45 and 46
ᮣ
As you can see, calculations involving binomial probabilities can become quite
extensive. Here again, a conceptual understanding of what the numbers mean can be
combined with the use of technology to solve significant applications of the idea. Most
graphing calculators provide a binomial probability distribution function, abbreviated
“binompdf(” and accessed using 2nd VARS (DISTR) 0:binompdf(. The function
requires three inputs: the number of trials n, the probability of success p for each trial, and
the value of k. As with the evaluation of other functions, k can be a single value or a list
of values enclosed in braces: “{ }.” The resulting calculation for Example 8 is shown in
Figure 9.67, and verifies each of the individual probabilities, although we must use the
right arrow to see them all. To find the sum of these probabilities, we simply precede the
“binompdf(” command with the “sum(” feature used previously. The final result is
shown in Figure 9.68, and verifies our earlier calculation. See Exercises 47 and 48.
Figure 9.67
Figure 9.68
E. You’ve just seen how
we can solve applications of
binomial powers
9.7 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
1. In any binomial expansion, there is always
more term than the power applied.
2. In all terms in the expanded form of 1a ϩ b2 n, the
exponents on a and b must sum to
.
3. To expand a binomial difference such as 1a Ϫ 2b2 5,
we rewrite the binomial as
and proceed
as before.
4. In a binomial experiment with n trials, the
probability there are exactly k successes is given
by the formula
.
5. Discuss why the expansion of 1a ϩ b2 n has n ϩ 1
terms.
6. For any defined binomial experiment, discuss the
relationships between the phrases, “exactly k
success,” and “at least k successes.”