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# Permutations

## The number of arrangements when order matters

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Permutations

Coach Coker is focused on building the defensive team. During games, the defensive team will need two players in the safety position to guard against long passes. To make sure the team has enough players who can play safety, Coach Coker wants to know how many different combinations of safety position players the team has if he has 6 players on the team who can play safety. The order of the arrangements is important because some of the players play other positions. How many combinations or permutations of two players are possible if order is important?

In this concept, you will learn about permutations and how to find all possible permutations.

### Guidance

A permutation is a combination where order makes a difference.

Look at the combinations listed in the chart below.

 SK KS DS JS SD KD DK JK SJ KJ DJ JD

There are 12 possible outcomes for this permutation because order is important and you do not to get rid of duplicates.

Using specific notation is an easier way to figure permutations than writing out all of the possibilities.

For example, in this chart there are initials for four boys in pairs - four taken two at a time. Here is how to write this as a permutation.

\begin{align*}P(4, 2)\end{align*}

This means there are four options taken two at a time.

To figure out the permutation, count down from 4 two numbers (4 and 3) then multiply them. There are two numbers to multiply because the boys were arranged two at a time.

\begin{align*}4 \times 3 = 12\end{align*}

There are 12 possible combinations. That is the same answer you found by writing things all out.

Here is another example.

How many ways can you arrange five swimmers in groups of three?

First, because you have groups of 3, multiply together the last 3 numbers in the count up to the number of items. Here is the permutation of 5 taken three at a time.

\begin{align*}P(5, 3) = 5 \times 4 \times 3\end{align*}

There are 60 possible combinations.

### Guided Practice

Evaluate \begin{align*}P(8, 3)\end{align*}

First, note the last three numbers leading up to 8. You use three because there are 3 to each group.

8, 7, 6

Next, multiply the three numbers to get the permutations.

\begin{align*}\begin{array}{rcl} P(8,3) &=& 8 \times 7 \times 6 \\ P(8,3) &=& 336 \end{array}\end{align*}

### Examples

Determine the following permutations.

#### Example 1

\begin{align*}P(9, 2)\end{align*}

First, note the last two numbers leading up to 9. You use two because there are 2 to each group.

9, 8

Next, multiply the numbers to get the permutations.

\begin{align*}\begin{array}{rcl} P(9,2) &=& 9 \times 8 \\ P(9,2) &=& 72 \end{array}\end{align*}

#### Example 2

\begin{align*}P(4, 3)\end{align*}

First, note the last three numbers leading up to 4. You use three because there are 3 to each group.

4, 3, 2

Next, multiply the three numbers to get the permutations.

\begin{align*}\begin{array}{rcl} P(4,3) &=& 4 \times 3 \times 2 \\ P(4,3) &=& 24 \end{array}\end{align*}

#### Example 3

\begin{align*}P(5, 2)\end{align*}

First, note the last two numbers leading up to 5. You use two because there are 2 to each group.

5, 4

Next, multiply the two numbers to get the permutations.

\begin{align*}\begin{array}{rcl} P(5, 2) &=& 5 \times 4\\ P(5, 2) &=& 20 \end{array}\end{align*}

Remember Coach Coker and his defensive team?

How many different combinations of 6 players in the 2 safety positions are there? When it comes to pairing the two players in the position, the order of the pairings is important.

First, express this as \begin{align*}P(6,2)\end{align*}, which means there are 5 players taken 2 at a time.

\begin{align*}P(6,2)\end{align*}

Next, note the last two numbers leading up to 6. You use two because there are 2 to each pairing.

6, 5

Then, multiply the two numbers to get the permutations.

\begin{align*}\begin{array}{rcl} P(6,2) &=& 6 \times 5 \\ P(6,2) &=& 30 \end{array}\end{align*}

There are 30 permutations.

### Explore More

Determine the following permutations.

1. \begin{align*}P(5, 2)\end{align*}
2. \begin{align*}P(6, 3)\end{align*}
3. \begin{align*}P(7, 2)\end{align*}
4. \begin{align*}P(5,4)\end{align*}
5. \begin{align*}P(7, 3)\end{align*}
6. \begin{align*}P(4, 4)\end{align*}
7. \begin{align*}P(5, 3)\end{align*}
8. \begin{align*}P(8, 4)\end{align*}
9. \begin{align*}P(9, 4)\end{align*}
10. \begin{align*}P(10, 3)\end{align*}
11. \begin{align*}P(12, 2)\end{align*}
12. \begin{align*}P(9, 3)\end{align*}
13. \begin{align*}P(8, 6)\end{align*}
14. \begin{align*}P(9, 3)\end{align*}
15. \begin{align*}P(10, 3)\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.19.

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
combination Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.
Outcome An outcome of a probability experiment is one possible end result.
Permutation A permutation is an arrangement of objects where order is important.
Probability Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.
Tree Diagram A tree diagram is a visual way of showing options and variables. The lines of a tree diagram look like branches on a tree.