12.2 Combinations and Binomial Theorem

p. 708

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 (objects arrangements are order unrelated, only the selected content is important)

A Combination is

A selection of r objects from a group of n distinct objects where order is not important

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

nCr can be read

“r combinations taken from a group of n objects” or “___________________”

• If there are 5 different colored marbles in a bag, how many ways can you pick a group of 2 of them?

=

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

52 5

52!

(52 5)!5!C

= 2,598,960

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:

4 1 13 5

4! 13!* * 5148

3!*1! 8!*5!C C

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

Examples using Combinations

In how many ways can you select 5 cards that are all Diamonds?

In how many ways can you select 2 cards that are the same suite?

In how many ways can you select 10 cards that are all the same suite?

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

• When finding the number of ways that an event A ___ B can occur, you _____

Deciding to + or *

• 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?

Suppose you can afford at most 3 ingredients

• How many different types can you order?• You can order an omelet w/ 0, or 1, or 2,

or 3 items and there are 10 items to choose from.

• Counting problems that involve ‘_________’ or ‘_________’ sometimes are easier to solve by subtracting possibilities you don’t want from the total number of possibilities.

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……

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

• So, from section 12.1 it would _________________________

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

• So:

Subtracting instead of adding:

• A summer concert series has 12 different artists performing. You want to attend at least 2 of them. How many different combinations of concerts can you attend?

Subtracting instead of adding:

• An amusement park has 20 different rides, and you want to go on at least 17 of them. How many different combinations of rides can you go on?

• Your favorite football team is playing a 10 game season. In how many ways can they end with 5 wins and 4 losses and 1 tie?

• 10 people go to dinner together. In how many different ways can you order 3 chicken dinners, 2 steak dinners, and 5 lobster dinners?

• There are 12 people on a committee, 8 women and 4 men. In how many ways can you elect 2 group leaders if you chose 1 man and 1 woman?

The Binomial Theorem

•0C0

•1C0 1C1

•2C0 2C1 2C2

•3C0 3C1 3C2 3C3

•4C0 4C1 4C2 4C3 4C4

• Etc…

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

• (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…

(a+b)n =

(a+b)2 =

(x-3)4 =

n

r

rrnrn baC

0

The Binomial Theorem

(a+3)5 =

•5C0a530+5C1a431+5C2a332+5C3a2b3+5

C4a134+5C5a035=

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

n

r

rrnrn baC

0

Assignment