In how many ways can the first, second, and third place winners be selected from 7 contestants?

Sam Kinsman

Hi Dorothy,

To distinguish between a permutation and a combination, you have to ask yourself whether the order matters. Think about whether the order the choose the items in makes a difference. If it does not, then the order does not matter, and it's a combination. If the order does matter, its a permutation.

In this case, there is a prize for 1st, 2nd, and 3rd place. That means that when the question says "In how many different ways can the judges award the 3 prizes?", the order matters, because if person A gets 1st place and B gets 2nd place, that's not the same as if A gets 2nd place and B gets 1st place.

When the question says "How many different groups of 3 people can get prizes?", it doesn't matter who gets which prize, it only matters who gets prizes and who doesn't. Therefore, the order doesn't matter, and it's a combination.

You may also find it helpful to take a look at this blog post:

https://magoosh.zendesk.com/hc/en-us/articles/115000372266-Combinations-vs-Permutation-Problem-GMAT-and-GRE-

May 14, 2018 • Reply

Permutations and Combinations

Permutations

When you have to take a smaller group of items from a larger group, you often need to know how many different ways there are of making a selection (this comes in handy when studying probability).

In some situations, the order of the items in the smaller group is important. For example, the diagram below shows all possible results for the top three dogs at a dog show:

Each arrangement lists the same dogs, but the order of first-, second-, and third-place winners differ. Arrangements such as these are called permutations. A permutation is an arrangement is which order is important. You can use the counting principle to count permutations.

Counting Permutations

Example 1

You have just downloaded 5 new songs. You can use the counting principle to count the different permutations of those 5 songs. This is the number of different sequences in which you can listen to the new songs on your playlist.

You can listen to the songs in 120 different orders.

Factorials!
In the last example, you evaluated 5 × 4 × 3 × 2 × 1. When you multiply a number by the number 1 less than itself, then by the number 1 less than that, and so on, all the way down to 1, this is known as a factorial. You can write "5 × 4 × 3 × 2 × 1" as "5!" which is read as "5 factorial."
5! = 5 × 4 × 3 × 2 × 1
n! = n × (n − 1) × (n − 2) × ... × 1
Note: The value of 0! is defined as 1.

Example 2

Twelve marching bands entered a competition. In how many different ways can first, second, and third places be awarded?

There are 1,320 different ways to award the three places.

Permutation Notation

The previous example shows how to find the number of permutations of 12 items taken 3 at a time. We can write this as 12P3. In general, the permutation formula is defined as follows:

The number of permutations of n objects taken r at a time = nPr =

Example:

In permutations, the order in which something is arranged is important.

A combination, on the other hand, is a group of items whose order is not important. For example, suppose you go to lunch with a friend. You choose milk, soup, and a salad. Your friend chooses soup, a salad, and milk. The order in which the items are chosen does not matter. You both have same meal.

Listing Combinations

Example

You have 4 tickets to the county fair and can take 3 of your friends. You can choose from Abby (A), Brian (B), Chloe (C), and David (D). How many different choices of groups of friends do you have?

Solution

List all possible arrangements of three friends. Then cross out any duplicate groupings that represent the same group of friends.

You have 4 different choices of groups to take to the fair.

Combination Notation

In Example 1, after you cross out the duplicate groupings, you are left with the number of combinations of 4 items chosen 3 at a time. Using notation, this is written 4C3.

To find the number of combinations of n objects taken r at a time, divide the number of permutations of n objects taken r at a time by r !.

Formula: Example:

Evaluating Combinations

Find the number of combinations if you select 3 items from a group of 8.

8C3

Find the number of combinations if you select 7 items from a group of 9.

9C7

Distinguishing Permutations and Combinations

Examples

State whether the possibilities can be counted using a permutation or combination. Then write an expression for the number of possibilities.

a. There are 8 swimmers in the 400 meter freestyle race. In how many ways can the swimmers finish first, second, and third?

Solution: Because the swimmers can finish first, second, or third, order is important. So the possibilities can be counted by evaluating 8P3.

b. Your track team has 6 runners available for the 4-person relay event. How many different 4-person teams can be chosen?

Solution: Order is not important in choosing the team members, so the possibilities can be counted by evaluating 6C4.

Practice

1. A(n) _?_ is a grouping of objects in which the order is not important.

2. A(n) _?_ is an arrangement of a group of objects in a particular order.

3. There are 12 members of a track team who want to run one of the legs in a 4 person relay race. Choose the calculation that you can use to find the number of ways that runners can be chosen for each of the legs of the relay race.

A.
B.
C.

4. You want to choose 3 different colors of balloons for a party. The balloons are available in 24 colors. How many ways can you choose 3 different colors of balloons?

5. There are 8 students participating in a car wash. How many ways can 2 of the students be chosen to hold signs advertising the car wash?

A. 8 B. 16
C. 28 D. 56

6. A bag contains 1 green marble, 1 blue marble, 1 red marble, and 1 white marble. How many ways can 3 marbles be randomly chosen from the bag, if the order in which the marbles are chosen is important?

Determine whether each situation below describes a permutation or combination. Then find the number of arrangements.

7. Ways to arrange the letters in the word GUITAR.

8. Ways to arrange 7 comic books on a shelf.

9. Ways to choose 4 different fish from 26 kinds of fish.

10. Ways to choose a president, vice-president, treasurer, and secretary from 18 members of a club.

11. Ways to choose 8 students to be extras in a play from 14 students.

12. Ways a coach can arrange the batting order of the 9 starting players of a baseball team

13. A door can be opened only with a security code that consists of five buttons: 1, 2, 3, 4, 5. A code consists of pressing any one button, or any two, or any three, or any four, or all five. How many possible codes are there? (You are to press all the buttons at once, so the order doesn't matter.)

14. How many different 3 digit numbers can you make using the digits 1, 4, 5, 6, 8, and 9, if no digit appears more than once in the number?

Answers

1. combination

2. permutation ("factorial" is also acceptable)

3. A

4. 2024

5. C

6. 24

7. permutation; 720

8. permutation; 5040

9. combination; 14,950

10. permutation; 73,440