5.7 The Binomial Theorem

1. Formulas for C(n,r)2. Binominal Coefficient3. Binomial Theorem4. Number of Subsets

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Formulas for C(n,r)

, 1 1,

! 1 1

!,

! !

, ,

P n r n n n rC n r

r r r

nC n r

r n r

C n r C n n r

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Example Routes

Work the route problem covered previously by selecting where in the string of length 7 the 4 E’s will be placed instead of the 3 S’s.Therefore the total number of possible routes is

7 6 5 47,4 35.

4 3 2 1C

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Notice that C(7,4) = C (7,3).

Binominal Coefficient

Another notation for C(n,r) is . n

r

n

r

4

is called a binominal coefficient.

Example Binominal Coefficient

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Binomial Theorem

Binomial Theorem

1 2 2

1

0 1 2

1

n n n n

n n

n n n

n n

n n

x y x x y x y

xy y

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Example Binominal Theorem

Expand (x + y )5.

5 5 5 5 5 5

0 1 2 3 4 51 5 10 10 5 1

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(x + y )5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

Number of Subsets

A set with n elements has 2n subsets.

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Example Number of Subsets

A pizza parlor offers a plain cheese pizza to which any number of six possible toppings can be added. How many different pizzas can be ordered?Ordering a pizza requires selecting a subset of the 6 possible toppings. There are 26 = 64 different pizzas.

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Summary Section 5.7 - Part 1

C(n,r) is also denoted by . The formula C(n,r) = C(n,n - r) simplifies the computation of C(n,r) when r is greater than n/2. The binomial theorem states that

1 2 2

1

0 1 2

1.

n n n n

n n

n n n

n n

n n

x y x x y x y

xy y

n

r

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Summary Section 5.7 - Part 2

A set with n elements has 2n subsets.

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