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10.2 Combinations and Binomial Theorem What you should learn: Goal Goal 1 1 Goal Goal 2 2 Use Combinations to count the number of ways an event can happen. Use the Binomial Theorem to expand a binomial that is raised to a power. 10.2 Combinatins and Binomial Theorem 10.2 Combinatins and Binomial Theorem

10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

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Page 1: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

10.2 Combinations and Binomial Theorem

What you should learn:

GoalGoal 11

GoalGoal 22

Use Combinations to count the number of ways an event can happen.

Use the Binomial Theorem to expand a binomial that is raised to a power.

10.2 Combinatins and Binomial Theorem10.2 Combinatins and Binomial Theorem

Page 2: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

In the last section we learned counting problems where

order was important• For other counting problems where

order is NOT important like cards, (the order you’re dealt is not important, after you get them, reordering them doesn’t change your hand)

• These unordered groupings are called Combinations

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 3: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

A Combination is a selection of r objects from a group of n objects

where order is not important

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 4: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Combination of n objects taken r at a time

• The number of combinations of r objects taken from a group of n distinct objects is

denoted by nCr and is:

!)!(

!

rrn

nCrn

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 5: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

For instance, the number of combinations of 2 objects taken from a group of 5 objects is

101*2*1*2*3

1*2*3*4*5

!2)!25(

!525

C

2

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 6: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Finding Combinations

• In a standard deck of 52 cards there are 4 suits with 13 of each suit.

• If the order isn’t important how many different 5-card hands are possible?

• The number of ways to draw 5 cards from 52 is

!5!*47

!47*48*49*50*51*52

!5)!552(

!52552

C

= 2,598,960

Page 7: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

In how many of these hands are all 5 cards the same suit?

• You need to choose 1 of the 4 suits and then 5 of the 13 cards in the suit.

• The number of possible hands are:

5148!5!*8

!8*9*10*11*12*13*

!1!*3

!3*4

!5!*8

!13*

!1!*3

!4* 51314 CC

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 8: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

How many 7 card hands are possible?

• How many of these hands have all 7 cards the same suit?

560,784,133!7!*45

!52752 C

6864* 71314 CC

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 9: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

When finding the number of ways both an event A and an event B can occur, you multiply.

When finding the number of ways that an event A OR B can occur, you +.

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 10: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Deciding to ADD or MULTIPLY

A restaurant serves omelets. They offer 6 vegetarian ingredients and 4 meat ingredients.

You want exactly 2 veg. ingredients and 1 meat. How many kinds of omelets can you order?

604*15!1!3

!4*

!2!4

!6* 1426 CC

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 11: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Suppose you can afford at most 3 ingredients

How many different types can you order?

You can order an omelet with 0, or 1, or 2, or 3 items and there are 10 items to choose from.

17612045101310210110010 CCCC

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 12: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Counting problems that involve ‘at least’ or ‘at most’ sometimes

are easier to solve by subtracting possibilities you

don’t want from the total number of possibilities.

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 13: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Subtracting instead of adding:

A theatre is having 12 plays. You want to attend at least 3. How many combinations of plays can you attend?

•You want to attend 3 or 4 or 5 or … or 12.

•From this section you would solve the problem using:

•Or……1212512412312 ... CCCC

Page 14: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

For each play you can attend you can go or not go.

•So, like section 10.1 it would be 2*2*2*2*2*2*2*2*2*2*2*2 =212

•And you will not attend 0, or 1, or 2.

•So:4017)66121(4096)(2 212112012

12 CCC

Page 15: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial
Page 16: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

0C0

1C0 1C1

2C0 2C1 2C2

3C0 3C1 3C2 3C3

4C0 4C1 4C2 4C3 4C4

Etc…

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 17: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Pascal's Triangle!

• 1• 1 1

• 1 2 1• 1 3 3 1

• 1 4 6 4 1• 1 5 10 10 5 1

• Etc…• This describes the coefficients in the

expansion of the binomial (a+b)n

Page 18: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

• (a+b)2 = a2 + 2ab + b2 (1 2 1)

• (a+b)3 = a3(b0)+3a2b1+3a1b2+b3(a0) (1 3 3 1)

• (a+b)4 = a4+4a3b+6a2b2+4ab3+b4 (1 4 6 4 1)

• In general…

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 19: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

(a+b)n (n is a positive integer)=

• nC0anb0 + nC1an-1b1 + nC2an-2b2 + …+ nCna0bn

• =

n

r

rrnrn baC

0

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 20: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

(a+3)5 =

• 5C0a530+5C1a431+5C2a332+5C3a233+

5C4a134+5C5a035=

• 1a5 + 15a4 + 90a3 + 270a2 + 405a + 243

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem

Page 21: 10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial

Assignment

12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem