4. The student council of 32 members must choose a president, a vice president, a secretary, and a treasurer. How many combinations of officers could there be?

3. How many different ways can 10 students line up for lunch?

1. Christian is going to dinner at Olive Garden. He is allowed to choose one of four pastas, one of three sauces, and one of three meats. How many different pasta dished could Christian make?

2. How would the number of combinations change if they were out of one of the sauces?

Let’s say there are 15 student council members named: Bruce, Shareeka, Yasmine, Jalen, Greg,

Stephon, Mikey, Bria, Joseph, Ekure, Anthony, Ar-Keno, Brittany, Tiffany, and Amber. If Shareeka, Tiffany, Yasmine, and Bruce are elected, would

the order in which they are chosen matter?

President Vice President Secretary Treasurer

Although the same individual students are listed in each example above, the listings are not the same. Each

listing indicates a different student holding each office. Therefore we must conclude that the order in which

they are chosen matters.

Shareeka Tiffany Yasmine Bruce

the same as…

Bruce Yasmine Tiffany Shareeka?

IS…

Permutation

Notation

Permutation

When deciding who goes 1st, 2nd, etc., order is important.

*Note  if  n = r   then   n Pr  =  n !

A permutation  is an arrangement or listing of objects in a specific order. 

The order of the arrangement is very important!! 

The notation for a permutation:       n Pr = n  is the total number of objects r is the number of objects selected (wanted)

!

( )!

n

n r

PermutationsSimplify each expression.

a. 12P2

b. 10P4

c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible?

12 • 11 = 132

10 • 9 • 8 • 7 = 5,040

= 20P4 = 20 • 19 • 18 • 17 = 116,280

Permutation Example:

Four runners are needed to run the 400 meter relay. How many different arrangements are there for the four runners to run the 4 legs

of the race?

The team has 8 sprinters. Four runners are needed to run the 400 meter relay. How

many different arrangements are there for the four runners, choosing from the 8 team

members, to run the 4 legs of the race?

24

1680

24

1.Bugs Bunny, King Tut, Kevin Jerome, and Daffy Duck are going to the movies (they are best friends). How many different ways can they sit in seats A, B, C, and D below?

2. Coach Hamilton is picking a captain and co-captain from her 15 players. How many possibilities does she have if they are all equally likely?

A B C D

210

Combinations

A selection of objects in which order is not important.

Example – 8 people pair up to do an assignment. How many different pairs are there?

CombinationsAB AC AD AE AF AG AH

BA BC BD BE BF BG BH

CA CB CD CE CF CG CH

DA DB DC DE DF DG DH

EA EB EC ED EF EG EH

FA FB FC FD FE FG FH

GA GB GC GD GE GF GH

HA HB HC HD HE HF HG

• The number of r-combinations of a set with n elements,

• where n is a positive integer and • r is a positive integer less than n, • i.e. the number of combinations of

r objects from n unlike objects is

!

! !n r

nC

r n r

Combinations

Example 1

How many different

ways are there to

select two class

representatives

from a class of 20

students?

!

! !n r

nC

r n r

Solution

• The answer is given by the number

of 2-combinations of a set with 20

elements.• The number of such combinations

is

20 2

20!190

2!18!C

Example 2

From a class of 24, Mrs. Shaffer is randomly selecting 3 to help Mrs. Benson with a project. How many

combinations are possible?

24 3

24!2024

3! 21!C

Your turn!For your school pictures, you can choose 4 backgrounds from a list of 10. How many combinations of backdrops are possible?

10 4

10!210

4! 6!C

Your turn!Coach Lynch randomly selects 3 people out of his class of 20 to go help him get ready for a lacrosse match. How many possibilities of people does he have?

20 3

20!1140

17! 3!C

To Sum it Up:

!

! !n r

nC

r n r

!

!n r

nP

n r

"My fruit salad is a combination of

apples, grapes and bananas"

the order doesn't matter

"The combination to

the safe is 472".

the order DOES matter

A Permutation is an ordered

Combination.

Both are counting principles

that tell you the total

number of possible

outcomes

Clarification on Combinations and Permutations

• "My fruit salad is a combination of apples, grapes and bananas"

We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

Clarification on Combinations and Permutations

• "The combination to the safe was 472".

Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.

To sum it up…

• If the order doesn't matter, it is a Combination.

• If the order does matter it is a Permutation.

A Permutation is an ordered Combination.

How many ways???

Fill in the blanks….

1. A permutation is an _________________ of objects in which order ___________ matter.

2. A combination is an _________________ of objects in which order ____________ matter.

homework

• Pg 344 1-18 even• Pg 349 1-10 all, and 17 - 18