Combination Calculator Online

A Combination Calculator is a free online tool to calculate the number of ways objects can be selected from a larger set. This uses the mathematical principle of combinations to calculate how many groups can be formed from a certain number of objects.

For example, if you had 10 students and wanted to select 3 for a project team, the Combination Calculator can give you the number of different teams that can be formed.

Combinations are used often in maths, probability, statistics, business analysis, research, and everyday life. Combinations are useful when choosing groups, committees, lottery numbers, teams, and calculating different outcomes.

Our Combination Calculator is simple to use. All you need to do is enter the total amount of items (n) and the amount you want to select (r). Then you will get the Total Number of combinations.

This can help students, teachers, researchers and professionals with combination problems.

How to use the Combination Calculator

The Combination Calculator is simple to use and just requires you to add 2 numbers.

Total (n) – Enter the total amount of items you have in your group. So if you had 12 people enter the number 12.

Select (r) – Enter how many items you would like to choose. If you had 4 people to choose enter the number 4.

Calculate – Press calculate.

Result – You will now see the Total Number of combinations.

Experiment with different numbers.

Ensure that the number you want to select (r) is less than or equal to the Total Number (n). Both numbers should be whole numbers and not negative.

Combination Calculator Formula

The formula used to calculate a combination is:

nCr = n! / [r! (n-r)!]

Where:

n = Total amount of items.

r = Amount of items you wish to select.

! = Factorials. This means multiplying the positive whole number by all the positive integers below it until you reach 1.

nCr = Total Number of combinations.

For example:

5! = 5 x 4 x 3 x 2 x 1 = 120

The combination formula calculates the number of unique selections. The order of the selection does not matter.

So selecting A,B,C is the same as selecting C,B,A.

Example

Let’s look at an example of how the Combination Calculator works.

Example

A company has 8 employees and they want to select a team of 3 for a special project. How many different teams can be formed?

We know:

Total Employees (n) = 8

Employees to select (r) = 3

We can use the combination formula:

nCr = n! / [r! (n-r)!]

Now let’s plug our numbers in:

8C3 = 8! / [3! (8-3)!]

8C3 = 8! / (3! x 5!)

8C3 = (8 x 7 x 6) / (3 x 2 x 1)

8C3 = 336 / 6

8C3 = 56

Answer = 56 combinations.

This means there are 56 possible combinations of teams that can be formed.

Understanding the Result

The result you get from a Combination Calculator shows how many unique groups can be formed.

You need to remember that combinations do not take into account the order.

For example, if you had to select 2 items from a group of A,B,C you would have:

A,B

A,C

B,C

This is 3 combinations in total.

Selecting A,B is the same as selecting B,A.

Useful Things to Note

The more items there are to choose from the more combinations there will be.

The more items you select the number of combinations can increase or decrease.

• If you select 0 items there will be 1 combination.
• If you select all the items there will be 1 combination.
• If you select 1 item from a group of n items there will be n combinations.

The result gives you the mathematical combinations. It does not give you the probability.

Things that Affect the Result

There are a few things that will affect the result.

1. Total Number of Items (n)

The Total Number of items will affect how many combinations there are. The bigger the group the more combinations there will be.

2. Number of Items to Select (r)

The number of items you want to select will play an important role in the combinations.

3. Order Doesn’t Matter

Combinations only take into account unique groups. If the order does matter then a permutation should be used.

4. No Repetition

The standard combination formula assumes that you can only select each item once.

5. Numbers Must Be Positive Whole Numbers

The calculator will only work with positive whole numbers. r must be less than or equal to n. If you enter invalid numbers then you cannot get a meaningful standard combination.

Benefits of using a Combination Calculator

A Combination Calculator can be a great tool to use for students, professionals and anyone that needs to solve a selection based maths problem.

Saves Time

Working out factorials can take a long time. Using the calculator will give you instant results.

Fewer Errors

As everything is automated there will be fewer errors when working out the combinations.

Helps Students and Teachers

Students can use the calculator to help them understand combinations, probability problems and check their answers to homework. Teachers can use the calculator to help teach maths concepts.

Helps With Selecting Teams

Companies, schools and organisations can work out how many different teams/committees can be formed from a group of people.

Useful for Probability Problems

Combinations are often used in probability and statistics. They can be used to calculate different selections and outcomes.

Helps With Research

Researchers and Analysts can use combinations when designing experiments, selecting samples and determining different groups.

Helps With Lotteries and Games

Combinations can be useful when trying to understand how many different number selections can be made in lotteries, card games etc.

Accuracy, Assumptions and Limitations

The Combination Calculator uses the standard combination formula to calculate combinations. This means it will give you accurate results.

However, there are some assumptions:

No repetition.

Order doesn’t matter.

Items are unique.

Numbers must be whole numbers and positive.

r must be less than or equal to n.

For example, if you were to select 3 different cards from a deck of cards this would be a standard combination as order doesn’t matter.

However, if you wanted to arrange those 3 cards in a specific order you would need to use a permutation.

Large numbers can cause the combination to be extremely large. Depending on the calculator and how precise it is with numbers, very large numbers may require specialist high precision calculations.

The calculator does not automatically calculate the probability. It will just give you the Total Number of selections. More information about probability would be required to know the likelihood of something happening.

FAQs

What is a Combination Calculator?

Combination Calculator is an online maths tool used to calculate the number of ways you can select a certain amount of items from a larger group. This uses the combination formula nCr = n! / [r! (n-r)!].

What is the difference between combinations and permutations?

Combinations are used when the order doesn’t matter. Permutations are used when the order does matter. For example, if you wanted to select 3 people for a team this would be a combination. However, if you wanted to assign them to 3 different roles this would be a permutation.

Can I calculate combinations if I select 0 items?

Yes. If you are selecting 0 items from any group there will only ever be 1 combination. This is because you are selecting nothing. Mathematically, nC0 = 1.

Can I select more items than I have?

No. When calculating a standard combination you cannot select more items than you have. For example, if you had 4 items and wanted to select 6 it wouldn’t be possible unless you changed the rules of selection.

Where are combinations used in real life?

Combinations are used when choosing teams, calculating lottery numbers, card games, probability, statistics, research, business planning, committees and much more. They are used when the order of selection doesn’t matter.