24

1.Bugs Bunny, King Tut, Caley Crouse, 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 Elkins is picking a captain and co-captain from his 15 players. How many possibilities does he have if they are all equally likely?

A B C D

210

Math I

UNIT QUESTION: How do you use probability to make plans and predict for the future?Standard: MM1D1-3

Today’s Question:What is the difference between a permutation and combination?Standard: MM1D1.b.

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 an integer with 0 <= r <= 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?

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, the teacher is randomly selecting 3 to help her 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!Venus Williams randomly selects 3 people out of her class of 20 to go to the courts and help her get ready for a tennis match. How many possibilities of people does she have?

20 3

20!1140

17! 3!C

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 is 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, then it is a Combination.

• Example: How many ways can we select three people from a group of 20?

• If the order does matter, then it is a Permutation.• Example: How many ways can we elect class

officers (pres., v.p., treas.) from a group of 20?A Permutation is an ordered Combination.

How many ways???