The number of combinations of

The formula is given by:

If you are not familiar with the *n*! (*n* factorial notation) then have a look the factorial lesson.

*Example: *

In how many ways can a coach choose three swimmers from among five swimmers?

*Solution: *

There are 5 swimmers to be taken 3 at a time.

Using the formula:

The coach can choose the swimmers in 10 ways.

*Example: *

Six friends want to play enough games of chess to be sure every one plays everyone else. How many games will they have to play?

*Solution: *

There are 6 players to be taken 2 at a time.

Using the formula:

They will need to play 15 games.*Example: *

In a lottery, each ticket has 5 one-digit numbers 0-9 on it.

a) You win if your ticket has the digits in any order. What are your changes of winning?

b) You would win only if your ticket has the digits in the required order. What are your chances of winning?

*Solution: *

There are 10 digits to be taken 5 at a time.

a) Using the formula:

The chances of winning are 1 out of 252.

b) Since the order matters, we should use permutation instead of combination.

P(10, 5) = 10 x 9 x 8 x 7 x 6 = 30240

The chances of winning are 1 out of 30240.

A combination is a grouping or subset of items. For a combination, the order does not matter.

How many committees of 3 can be formed from a group of 4 students?

This is a combination and can be written as C(4,3) or

Examples:

1. The soccer team has 20 players. There are always 11 players on the field. How many different groups of players can be on the field at any one time?

2. A student need 8 more classes to complete her degree. If she met the prerequisites for all the courses, how many ways can she take 4 classes next semester?

3. There are 4 men and 5 women in a small office. The customer wants a site visit from a group of 2 man and 2 women. How many different groups can be formed from the office?

Example:

A bucket contains the following marbles: 4 red, 3 blue, 4 green, and 3 yellow making 14 total marbles. Each marble is labeled with a number so they can be distinguished.

1. How many sets/groups of 4 marbles are possible?

2. How many sets/groups of 4 are there such that each one is a different color?

3. How many sets of 4 are there in which at least 2 are red?

4. How many sets of 4 are there in which none are red, but at least one is green?

Examples:

1. A museum has 7 paintings by Picasso and wants to arrange 3 of them on the same wall. Ho many ways are there to do this?

2. How many ways can you arrange the letters in the word LOLLIPOP?

3. A person playing poker is dealt 5 cards. How many different hands could the player have been dealt?

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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