# 1.5: Mixed Combinations and Permutations

**At Grade**Created by: Bruce DeWItt

### Learning Objectives

- Determine whether a situation involves permutations or combinations
- Understand the mathematical implications of the words 'and' & 'or'

Having covered the basics of combinations and permutations, you are ready to have a mixture of problems with slight variations. A common variation involves an understanding of some key words used in mathematics. Commonly, the word "**and**" indicates multiplication and the word "**or**" indicates addition. Consider the examples below.

#### Example 1

In how many ways can committee of 3 people be chosen if there are 8 men and 4 women available for selection and we require that two men and one woman be on the committee?

#### Solution

The order that we place the people on a committee does not matter. It makes no difference if you are the first person or the last person selected for the committee. Either you are on the committee or you are not on the committee, therefore this is a combination question. Notice that we want two men

andone women. The word 'and' indicates multiplication. In other words, we will look for the product of how many ways we can select two men from eight and one women from four. \begin{align*}_8 C _2\times_4 C _1=28\times4=112\end{align*}8C2×4C1=28×4=112 . There are 112 ways to select this committee of 3 people.

#### Example 2

In how many ways can a committee of 5 people be chosen if there are 7 men and 5 women available for selection and we require *at least* 4 women on the committee?

#### Solution

We first ask "Does order matter?". In this case, the order that someone is placed on a committee does not matter. Either you are on the committee or you are not. Once again, we are dealing with a combination question. The key phrase in this example is

at least. This can be interpreted to mean that we either select 4 womenand1 manor5 womenand0 men.

Remember that the word 'and' indicates multiplication and the word 'or' indicates addition. It looks like we are going to have some addition and some multiplication in this problem.

\begin{align*}_5C_4\times_7C_1+_5C_5\times_7C_0=5\times7+1\times1=35+1=36\end{align*}

5C4×7C1+5C5×7C0=5×7+1×1=35+1=36 . There are 36 ways to put this committee together.

#### Example 3

In a certain country, there are two political parties. Each party is responsible for nominating both a presidential and vice-presidential candidate. The candidates will participate in a debate once they are chosen. In the first party, there are 6 candidates available and in the second party there are 5 candidates available. How many different debate combinations are possible?

#### Solution

The order that we select the candidates does make a difference. Selecting party member 'A' for a presidential candidate and party member 'B' for a vice-presidential candidate is different than selecting party member 'B' for a presidential candidate and party member 'A' for a vice-presidential candidate. Therefore, this is a permutations question. Since we will select candidates from the first party

andcandidates from the second party, we expect there to be multiplication in this problem as well.

\begin{align*}_6P_2\times_5P_2=30\times20=600\end{align*}

6P2×5P2=30×20=600 . There are 600 different ways that the debate participants can be chosen.

### Problem Set 1.5

#### Exercises

1) In your own words, state how you can tell the difference between a combination and permutation problem.

2) Your closet contains 10 different styles of shoes. In how many ways can you pick out five different styles of shoes for the school week if you don't care which day of the week you wear each style?

3) Your closet contains 10 different styles of shoes. In how many ways can you pick out five different styles of shoes for the school week if you do care which day of the week you wear each style?

4) You are drawing a rainbow using five different colored crayons from your box of 24 colors. In how many ways can you draw a rainbow if the first color you pick will be the top layer and so on?

5) You are drawing a rainbow using five different colored crayons from your box of 24 colors. In how many ways can you pick the five colors for your rainbow?

6) Suppose 5 cards are dealt from a standard deck of 52 cards.

a) How many unique 5-card hands are possible?

b) In how many different orders can 5 cards be dealt from a standard deck?

7) Suppose the majority party in a foreign country must select a prime minister and secretary of state from an eligible group of 36 party members. In how many ways can this be done?

8) There are 7 women and 5 men in a department. Four people are needed for a committee.

a) In how many ways can a committee of 4 people be selected?

b) In how many ways can this committee be selected if there must be

exactly2 men and 2 women on the committee?

c) In how many ways can this committee be selected if there must be

at least2 women on the committee?

9) A company has 8 cars and 11 trucks. The state inspector will select 3 cars and 4 trucks to be tested for safety inspections. In how many ways can this be done?

10) In a train yard there are 4 tanker cars, 12 boxcars, and 7 flatcars available for a train. In how many ways can a train be made up consisting of 2 tanker cars, 5 boxcars, and 3 flatcars?

11) Flakes-R-Us cereal comes in two types, Sugar Sweet and Touch O' Honey. If a researcher has ten boxes of each type, how many ways can she select two boxes of each for a quality control test?

12) In how many ways can a jury of 12 people be selected from a pool of 12 men and 10 women?

13) In how many ways can a jury of 6 men and 6 women be selected from a pool of 12 men and 10 women?

14) A corporation president must select a manager and assistant manager for each of two stores. In how many ways can this be done if the first store has 9 employees and the second store has 7 employees? (Employees will stay at their current stores.)

15) Suppose that in this trimester that every sophomore is required to take 2 math classes, 2 social studies classes, and a reading class. How many different combinations of teachers are possible for a given student if there are 9 math teachers, 12 social studies teachers, and 4 reading teachers available? (No student will have the same teacher for two different hours.)

16) In how many different ways can six people be assigned to three offices if there will be two people in each office?

#### Review Exercises

17) Use the formula for combinations to find the value of \begin{align*}_7C_3\end{align*}

18) Use the formula for permutations to find the value of \begin{align*}_6P_4\end{align*}

19) In how many ways can the letters in the word 'magic' be arranged?

20) How many different outcomes are possible for the total of when two 4-sided dice are rolled?

21) A teacher will select three students to work problems on the board from her class of 34 students. In how many ways can this be done if the three problems to be worked are #11, #14, and #26?

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