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Permutations

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Permutations

Have you ever created an event at school? Look at this collaboration.

Greg and Joanna are in charge of creating an order for the Talent Show. There are 6 sixth graders, 10 seventh graders and 6 eighth graders who have entered the contest.

The order that the students perform in makes a difference, and the sixth graders will all perform together. Then the seventh graders will perform and finally the eighth graders will perform.

Greg and Joanna begin with the sixth graders. Since there are six sixth graders who are performing and order does make a difference, how many different arrangements of the order are possible?

“I don’t have a clue how to figure this out,” Joanna says to Greg.

“I think I do, let me get a piece of paper.”

While Greg gets a piece of paper, it is time for you to think about this problem. Solving it has to do with something called a “permutation.” This lesson is all about permutations and how to figure them out.

Pay attention and at the end of the Concept you will know how to help Greg and Joanna figure out the order of sixth graders.

Guidance

You can make all kinds of combinations. Let’s say that you are making a pizza with pepperoni, mushrooms and peppers. It doesn’t matter which order you put the toppings on the pizza. You will still have the same pizza.

Sometimes, order does make a difference. When you arrange different objects or events and order is important, we call this a permutation.

This Concept is all about permutations.

Think about spelling.

Consider the word CAT. Clearly, order is important when you spell a word. You can write all of the correct letters, but if you don’t put them in the correct order, you don’t spell CAT. For example, here are some orders of C, A, and T that don’t spell CAT;

& \text{Incorrect orders:} &&  ACT, ATC, CTA, TAC, TCA\\& \text{Correct orders:} && CAT

This is a situation where a permutation makes a difference. We can also use permutations to solve problems. To use permutations to solve problems, you need to be able to identify the problems in which order, or the arrangement of items, matters.

Tomás wants to know how many 3-digit numbers he can write using the digits 7, 8, and 9 without repeating any of the digits. Does order matter for this problem?

Step 1 : Write out a single order.

789

Step 2 : Now rearrange the order. Did you change the outcome? If so, then order matters.

798

This is different from the original number

Each arrangement of digits is a different permutation.

The softball coach needs to determine how many different batting lineups she can make out of her first three batters, Able, Baker, and Chan. Does order matter for this problem?

Step 1 : Write out a single lineup.

Able, Baker, Chan

Step 2 : Now rearrange the lineup. Did you change the outcome? If so, then order matters.

Able, Chan, Baker–this order is different from the original

Each arrangement of batters is a different permutation.

Write whether or not order is important for each scenario and why.

Example A

How many 3-letter words can Brenda write using the letters E, T, A without repeating letters?

Solution: Order is important

Example B

Five different cars entered the race painted the following colors: red, orange, blue, white, purple? In how many different ways can the cars finish the race?

Solution: Order is important

Example C

At the breakfast buffet you can take any three of the following: eggs, pancakes, potatoes, cereal, waffles. How many different 3-item breakfasts can you get?

Solution: Order is not important

Here is the original problem once again. Reread it and then work through figuring out the permutation.

Greg and Joanna are in charge of creating an order for the Talent Show. There are 6 sixth graders, 10 seventh graders and 6 eighth graders who have entered the contest.

The order that the students perform in makes a difference, and the sixth graders will all perform together. Then the seventh graders will perform and finally the eighth graders will perform.

Greg and Joanna begin with the sixth graders. Since there are six sixth graders who are performing and order does make a difference, how many different arrangements of the order are possible?

“I don’t have a clue how to figure this out,” Joanna says to Greg.

“I think I do, let me get a piece of paper.”

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

6 sixth graders

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,” Greg 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. Greg and Joanna 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

Permutation
a combination where the order matters.

Guided Practice

Here is one for you to try on your own.

Henry has a combination lock. He can program the combination of his lock using the numbers 1, 5 and 18. Henry can only choose one arrangement of the numbers.

Is this a permutation? Why?

Answer

Henry and his combination lock is an example of a permutation because there is only one way to order the numbers and order does make a difference.

Video Review

This is a James Sousa video on permutations.

Practice

Directions: Decide whether or not order matters for each of the following scenarios. Briefly explain your reasoning.

1. Doug is going to use the following 5 letters to create his new 3-letter computer password: B, F, G, L, and T. How many different passwords can he create if he doesn’t repeat any letters?

2. Violin players in the orchestra include Jerry, Kerry, Barry, Mary, Sherry, Harry, Terri, and Perry. How many different 3-person trios can you make?

3. The 3 different numbers for Arun’s lock are 14, 35, and 20. How many different combinations must Arun try before he’ll be sure he can open his lock?

4. Mr. Chen has decided that he’s going to give Nikki, Mickey, and Hickey awards for the essay contest. What he doesn’t know is who will get 1st prize, 2nd prize, and 3rd prize. How many different ways can Mr. Chen give out the prizes?

5. Five candidates are running for 2 student senate seats: Bo, Jo, Mo, Zo, and Ro. How many different pairs of senators can there be?

6. Five skaters are competing in the County Championship Finals: Miller, Diller, Hiller, Giller, and Stiller? How many different ways can they finish first, second, third, fourth, and fifth?

Directions: Find the number of permutations for each problem.

7. How many 3-letter words can Brenda write using the letters A, B, and C without repeating any of the letters?

8. How many 4-digit numbers can Brenda write using the digits 1, 3, 5, 7 without repeating any of the digits?

9. Doug, Eileen, Francesca, and Garth all entered the swimming race. In how many different orders can the four racers finish?

10. Miguel is serving soup, salad, pasta, and fish for dinner. In how many different orders can he serve the four dishes?

11. Mike has 4 different playing cards: Ace, King, Jack, and Ten. How many different 4-card arrangements can he make?

12. Marlena strung 5 charms on a bracelet–a star, a fish, a diamond, a moon, and a baby shoe. Into how many different orders can she arrange the 5 charms?

13. Michelle forgot her 6-letter computer password. She knows she used the letters H, I, J, K, L, and M in the password and that she didn’t repeat any of the letters. How many different passwords must she try before she is sure to hit the correct one?

14. Seven skaters are competing in the County Championship Finals. In how many different orders can they finish first, second, third, fourth, fifth, sixth, and seventh?

15. If there are eight skaters competing, how does this change the number of permutations?

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