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# Permutations

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Have you ever tried to organize people for an event? Well, there is a talent show at the amusement park. Take a look.

Kyle and Taylor are in charge of creating an order for the Talent Show. There are 6 boys, 10 girls and 6 adults who have entered the show.

The order that the students perform in makes a difference, and the boys will all perform together. Then the girls will perform and finally the adults will perform.

Kyle and Taylor begin with the boys Since there are six boys who are performing and order does make a difference, how many different arrangements of the order are possible?

Solving this dilemma has to do with something called a “permutation.” This Concept is all about permutations and how to figure them out.

### Guidance

A permutation is a combination where order makes a difference. In the last section, we didn’t care about order. We just cared about the pairings.

If we had cared about order, then SK and KS would be two different things.

We would have counted ALL of the possible combinations and they would be included in our permutation because order matters.

Let’s look at the permutations from the last problem.

$&\text{SK} && \text{KS} && \text{DS} && \text{JS}\\&\text{SD} && \text{KD} && \text{DK} && \text{JK}\\ &\text{SJ} && \text{KJ} && \text{DJ} && \text{JD}$

Here we have 12 possible outcomes for this permutation.

Is there any easier way to figure this out besides writing out all of the possibilities?

Yes there is. In fact, there is a way to do this using specific notation.

First, we had four boys in pairs. Four taken two at a time, here is our permutation.

$P(4,2)$

This tells us that we have four options taken two at a time.

We figure out the permutation by counting down from four two numbers and we multiply them.

$4 \cdot 3$

Notice that we multiply the last two digits in the count up to four. There are two numbers to multiply because the boys were arranged two at a time. Next we multiply.

4 $\times$ 3 $=$ 12

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

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

This time we have groups of 3, so we multiply together the last 3 numbers in the count up to our number of items. Here is the permutation of 5 taken three at a time.

$P(5, 3) = 5 \times 4 \times 3$

There are 60 possible combinations.

Practice figuring out the following permutations.

#### Example A

$P(9, 2)$

Solution: $72$

#### Example B

$P(4, 3)$

Solution: $24$

#### Example C

$P(5, 2)$

Solution: $20$

Now back to the original problem.

Kyle and Taylor are in charge of creating an order for the Talent Show. There are 6 boys, 10 girls and 6 adults who have entered the show.

The order that the students perform in makes a difference, and the boys will all perform together. Then the girls will perform and finally the adults will perform.

Kyle and Taylor begin with the boys Since there are six boys who are performing and order does make a difference, how many different arrangements of the order are possible?

Kyle takes out a piece of paper and writes this on it.

6 boys

$6 \times 5 \times 4 \times 3 \times 2 \times 1 = the \ number \ of \ possible \ arrangements$

“You see, it makes a difference, so we can use a factorial,” Kyle explains. “Now we will know how many possible arrangements of sixth graders there are.”

$6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

There are 720 possible combinations. Kyle and Taylor probably need to narrow this down a little further because that is a lot of possible arrangements. They decide to make singers one category. That will help them with possible combinations.

### Vocabulary

Here are the vocabulary words in this Concept.

Probability
the chances or likelihood that an event will happen.
Outcome
the end result
Tree Diagram
a visual way of showing options and variables in an organized way. The lines of a tree diagram look like branches on a tree.
Combination
an arrangement of options where order does not make a difference.
Permutation
an arrangement of options where order does make a difference.

### Guided Practice

Here is one for you to try on your own.

Evaluate $P(8,3)$

This means that we have eight items taken three at a time. Here is how we can write out this problem.

$8 \times 7 \times 6$

### Video Review

Here are videos for review.

### Practice

Directions: Figure out the following permutations.

1. $P(5,2)$

2. $P(6, 3)$

3. $P(7, 2)$

4. $P(5, 4)$

5. $P(7, 3)$

6. $P(4, 4)$

7. $P(5, 3)$

8. $P(8, 4)$

9. $P(9, 4)$

10. $P(10, 3)$

11. $P(12, 2)$

12. $P(9, 3)$

13. $P(8, 6)$

14. $P(9,3)$

15. $P(10,3)$