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B. Binomial Coefficients and Factorials

# 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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Section 9.7 The Binomial Theorem

Solution

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.”

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