7.4 - 1 Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Row 0 1 2 3 4 5.

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Pascal’s Triangle

1 1 1

1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Row012345

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Pascal’s Triangle

With the coefficients arranged in this way, each number in the triangle is the sum of the two numbers directly above it (one to the right and one to the left).For example, in row four, 1 is the sum of 1 (the only number above it), 4 is the sum of 1 and 3, 6 is the sum of 3 and 3, and so on. This triangular array of numbers is called Pascal’s triangle, in honor of the seventeenth-century mathematicianBlaise Pascal. It was, however, known long before his time.

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Pascal’s Triangle

To find the coefficients for (x + y)6 we need to include row six in Pascal’s triangle. Adding adjacent numbers, we find that row six is

1 6 15 20 15 6 1.

Using these coefficients, we obtain the expansion of (x + y)6 :

6 6 5 4 2 3 3 2 4 5 6( ) 6 15 20 15 6 .x y x x y x y x y x y xy y

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n-Factorial

Although it is possible to use Pascal’s triangle to find the coefficients of (x + y)n for any positive integer n, this calculation becomes impractical for large values of n because of the need to write all the preceding rows. A more efficient way of finding these coefficients uses factorial notation. The number n! (read “n-factorial”) is defined as shown in the next slide.

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n-Factorial

For any positive integer n,

! ( 1)( 2) (3)(2)(1) and 0! 1.n n n n

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Binomial Coefficients

Now look at the coefficients of the expansion

5 5 4 3 2 2 3 4 5( ) 5 10 10 5 .x y x x y x y x y xy y

The coefficient of the second term, 5x4y, is 5, and the exponents on the variablesare 4 and 1. Note that

5!

5 .4!1!

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Binomial Coefficient

For nonnegative integers n and r, withr ≤ n,

!.

!( )!n r

n nC

r n rr

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Binomial Coefficients

The binomial coefficients are numbers from

Pascal’s triangle. For example, is the first

number in row three, and is the fifth number in row seven.Graphing calculators are capable of finding binomial coefficients. A calculator with a 10-digit display will give exact values for n! for n ≤ 13 and approximate values of n! for 14 ≤ n ≤ 69.

3

0

7

4

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Example 1EVALUATING BINOMIAL COEFFICIENTS

Solution

d.

Evaluate the binomial coefficient.

12 10

12! 12!66

10!(12 10)! 10!2!C

12 10C

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The Binomial TheoremOur observations about the expansion of (x + y)n are summarized as follows.

1. There are n + 1 terms in the expansion.2. The first term is xn, and the last term is yn.3. In each succeeding term, the exponent on x decreases by 1 and the exponent on y increases by 1.4. The sum of the exponents on x and y in any term is n.5. The coefficient of the term with xryn – r or xn – r y r is

.n

r

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Binomial Theorem

For any positive integer n and any complex numbers x and y,

1 2 2 3 3( )

1 2 3n n n n nn n n

x y x x y x y x y

1 .1

n r n nn nx y xy y

r n

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Note The binomial theorem looks much more manageable written as a series. The theorem can be summarized as

0

( ) .n

n n r r

r

nx y x y

r

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Example 2APPLYNG THE BINOMIAL THEOREM

Write the binomial expansion of (x + y)9.

Solution

9 9 8 7 2 6 3 5 4 4 5

3 6 2 7 8 9

9 9 9 9 9( )

1 2 3 4 5

9 9 9.

6 7 8

x y x x y x y x y x y x y

x y x y xy y

Now evaluate each of the binomial coefficients.

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Example 2APPLYNG THE BINOMIAL THEOREM

9 9 8 7 2 6 3 5 4 4 5

3 6 2 7 8 9

9! 9! 9! 9! 9!( )

1!8! 2!7! 3!6! 4!5! 5!4!9! 9! 9!

6!3! 7!2! 8!1!

x y x x y x y x y x y x y

x y x y xy y

9 8 7 2 6 3 5 4 4 5

3 6 2 7 8 9

9 36 84 126 126

84 36 9

x x y x y x y x y x y

x y x y xy y

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kth Term of the Binomial Expansion

The kth term of the binomial expansion of (x + y)n, where n ≥ k – 1, is

( 1) 1.1

n k knx y

k

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kth Term of the Binomial Expansion

To find the kth term of the binomial expansion, use the following steps.

Step 1 Find k – 1.This is the exponent on the second part of the binomial.

Step 2 Subtract the exponent found in Step 1 from n to get the exponent on the first part of the binomial.

Step 3 Determine the coefficient by using the exponents found in the first two steps and n.

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Example 5FINDING A PARTICULAR TERM OF A BINOMIAL EXPANSION

Find the seventh term of (a + 2b)10.

Solution

In the seventh term, 2b has an exponent of 6 while a has an exponent of 10 – 6 or 4. The seventh term is

4 6 4 6 4 610(2 ) 210 (64 ) 13,440 .

6a b a b a b