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6.3. Find Probabilities Using Combinations. Count combinations. Example 1. Count the combination of two letters from the list A, B, C, D, E. Solution.

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6.3

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6.3

Find Probabilities Using Combinations

Count combinations

Example 1

Count the combination of two letters from the list A, B, C, D, E.

Solution

List all the permutations of two letters in the list A,B, C, D, E. Because order is important in a combination, cross out any duplicate pairs.

AB

AC

BA

BD

BE

CA

AE

BC

CB

CD

DA

CE

DC

EA

EB

EC

DB

DE

ED

There are ___ possible combinations of 2 letters from the list A, B, C, D, E.

10

6.3

Find Probabilities Using Combinations

The number of combinations of n objects taken r at a time, where r < n, is given by:

COMBINATIONS

Formula

Example

The number of combinations of 5 objects taken 2 at a time is:

6.3

Find Probabilities Using Combinations

Use the combinations formula

Example 2

Toppings You order a pizza at a restaurant. You can choose 3 toppings from a list of 12. How many combinations of toppings are possible?

Solution

The order in which you choose the toppings is not important. So, to find the number of combinations of 12 toppings taken 3 at a time, find 12C3.

220

There are _____ different combinations of toppings.

6.3

Find Probabilities Using Combinations

Checkpoint. Complete the following exercises.

• Count the combination of two letters from the list A, B, C, D, E, F.

6.3

Find Probabilities Using Combinations

Checkpoint. Complete the following exercises.

• In Example 2, suppose you can choose only 2 toppings out of 12 topping choices. How many combinations are possible?

6.3

Find Probabilities Using Combinations

Find a probability using combinations

Example 3

Scholarships A committee must award three students with scholarships. Fifteen students are candidates for the scholarship including you and your two best friends. If the awardees are selected randomly, what is the probability that you and your two best friends are awarded the scholarships?

Solution

Write the number of possible outcomes as the number of combinations of 15 candidates taken 3 at a time, 15C3.

Step 1

6.3

Find Probabilities Using Combinations

Find a probability using combinations

Example 3

Scholarships A committee must award three students with scholarships. Fifteen students are candidates for the scholarship including you and your two best friends. If the awardees are selected randomly, what is the probability that you and your two best friends are awarded the scholarships?

Solution

one

Step 2

Find the number of favorable outcomes. Only ____ of the possible combinations includes scholarships for you and your two best friends.

Step 3

Calculate the probability.

P(scholarships awarded to you and your friends) = _____

6.3

Find Probabilities Using Combinations

Checkpoint. Complete the following exercises.

• In Example 3, suppose there are 20 candidates for the scholarships. Find the probability that you and your two best friends are awarded the 3 scholarships.

P(scholarships awarded to you and your friends) =

6.3

Find Probabilities Using Combinations

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