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Math in Our World. Section 11.2. Combinations. Learning Objectives. Distinguish between combinations and permutations. Find the number of combinations of n objects taken r at a time. Use the combination rule in conjunction with the fundamental counting principle. Combinations.

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Section 11.2


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Presentation Transcript
learning objectives
Learning Objectives
  • Distinguish between combinations and permutations.
  • Find the number of combinations of n objects taken r at a time.
  • Use the combination rule in conjunction with the fundamental counting principle.
combinations
Combinations

A selection of objects without regard to order is called a combination.

example 1 comparing permutations and combinations
EXAMPLE 1 Comparing Permutations and Combinations

Given four housemates, Ruth, Elaine, Ama, and Jasmine, list the permutations and combinations when you are selecting two of them.

SOLUTION

We’ll start with permutations, then eliminate those that have the same two people listed.

example 1 comparing permutations and combinations1
EXAMPLE 1 Comparing Permutations and Combinations

SOLUTION

Permutations

Ruth Elaine Elaine Ruth RuthAma

Ama Ruth Ruth Jasmine Jasmine Ruth

Elaine Ama Ama Elaine Elaine Jasmine

Jasmine Elaine Ama Jasmine Jasmine Ama

Combinations

Ruth Elaine Ruth Ama Ruth Jasmine

Elaine Ama Elaine Jasmine Ama Jasmine

There are 12 permutations, but only 6 combinations.

example 2 identifying permutations and combinations
EXAMPLE 2 Identifying Permutations and Combinations

Decide if each selection is a permutation or a combination.

(a) From a class of 25 students, a group of 5 is chosen to give a presentation.

(b) A starting pitcher and catcher are picked from a 12-person intramural softball team.

example 2 identifying permutations and combinations1
EXAMPLE 2 Identifying Permutations and Combinations

Decide if each selection is a permutation or a combination.

(a) From a class of 25 students, a group of 5 is chosen to give a presentation.

(b) A starting pitcher and catcher are picked from a 12-person intramural softball team.

SOLUTION

(a) This is a combination because there are no distinct roles for the 5 group members, so order is not important.

(b) This is a permutation because each selected person has a distinct position, so order matters.

the combination rule
The Combination Rule

The number of combinations of n objects taken r at a time is denoted by nCr , and is given by the formula

example 3 using the combination rule
EXAMPLE 3 Using the Combination Rule

How many combinations of four objects are there taken two at a time?

SOLUTION

Since this is a combination problem, the answer is

This matches our result from Example 1.

example 3 using the combination rule1
EXAMPLE 3 Using the Combination Rule

How many combinations of four objects are there taken two at a time?

example 4 an application of combinations
EXAMPLE 4 An Application of Combinations

While studying abroad one semester, Tran is required to visit 10 different cities. He plans to visit 3 of the 10 over a long weekend. How many different ways can he choose the 3 to visit? Assume that distance is not a factor.

example 4 an application of combinations1
EXAMPLE 4 An Application of Combinations

While studying abroad one semester, Tran is required to visit 10 different cities. He plans to visit 3 of the 10 over a long weekend. How many different ways can he choose the 3 to visit? Assume that distance is not a factor.

SOLUTION

The problem doesn’t say anything about the order in which they’ll be visited, so this is a combination problem.

example 5 choosing a committee
EXAMPLE 5 Choosing a Committee

At one school, the student government consists of seven women and five men. How many different committees can be chosen with three women and two men?

example 5 choosing a committee1
EXAMPLE 5 Choosing a Committee

At one school, the student government consists of seven women and five men. How many different committees can be chosen with three women and two men?

SOLUTION

First, we will choose three women from the seven candidates. This can be done in 7C3= 35 ways.

Then we will choose two men from the five candidates in 5C2= 10 ways.

Using the fundamental counting principle, there are 35 x 10 = 350 possible committees.

example 6 designing a calendar
EXAMPLE 6 Designing a Calendar

To raise money for a charity event, a sorority plans to sell a calendar featuring tasteful pictures of some of the more attractive professors on campus. They will need to choose six models from a pool of finalists that includes nine women and six men. How many possible choices are there if they want to feature at least four women?

example 6 designing a calendar1
EXAMPLE 6 Designing a Calendar

SOLUTION

Since we need to include at least four women, there are three possible compositions: four women and two men, five women and one man, or six women and no men.

Four women and two men:

Using Fundamental Counting Principle:

number of ways to choose 4 women times the number of ways to choose 2 men

example 6 designing a calendar2
EXAMPLE 6 Designing a Calendar

SOLUTION

Five women and one man:

Six women and no men:

The total number of possibilities is 1,890 + 756 + 84 = 2,730.