**Instructions:**

- Enter values for 'n' and 'r' in the respective fields.
- Click "Calculate" to compute the result (nCr).
- Your detailed calculation and explanation will be displayed below.
- The calculation history will also appear below.
- Use "Clear" to reset the input fields and result.
- Click "Copy Result" to copy the result to the clipboard.

**Detailed Calculation**

**Calculation History**

A combinations calculator is a tool that allows users to calculate the number of combinations of a given set of items. A combination is a subset of a set of items in which the order of the items does not matter.

**Concepts**

The following are some of the key concepts that underlie combinations calculators:

**Set:**A set is a collection of distinct objects.**Subset:**A subset of a set is a collection of objects that are members of the original set.**Combination:**A combination is a subset of a set in which the order of the items does not matter.

**Formulae**

The following formula is used to calculate the number of combinations of a given set of items:

```
nCr = n! / r!(n - r)!
```

where:

- n is the number of items in the set
- r is the number of items in the combination

For example, if you have a set of 5 items and you want to calculate the number of combinations of 3 items, you would use the following formula:

```
5C3 = 5! / 3!(5 - 3)! = 10
```

Therefore, there are 10 combinations of 3 items from a set of 5 items.

**Benefits of using a combinations calculator**

There are several benefits to using a combinations calculator, including:

**Accuracy:**Combinations calculators are very accurate, as they use sophisticated mathematical algorithms to perform their calculations.**Convenience:**Combinations calculators can save users a lot of time and effort, as they can perform complex calculations quickly and easily.**Flexibility:**Combinations calculators can be used to calculate the number of combinations of any set of items, regardless of the size of the set.**Versatility:**Combinations calculators can be used in a variety of fields, including mathematics, computer science, and probability.

**Interesting facts about combinations**

- The number of combinations of a set of items is always greater than or equal to the number of permutations of the same set of items.
- The number of combinations of a set of items is equal to the number of ways to choose the order of the items in the set and then divide by the number of times that each order is counted.
- The number of combinations of a set of items can be used to calculate the probability of certain events, such as the probability of getting a certain number of heads on a coin toss.

**Scholarly references**

**Kenneth H. Rosen:**Discrete Mathematics and Its Applications, 8th Edition, McGraw-Hill Education, 2019**Susan S. Epp:**Discrete Mathematics with Applications, 5th Edition, Cengage Learning, 2018**Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein:**Introduction to Algorithms, 3rd Edition, MIT Press, 2009

**Conclusion**

Combinations calculators are a valuable tool for anyone who needs to calculate the number of combinations of a given set of items. They are accurate, convenient, flexible, and versatile. Combinations calculators can be used in a variety of fields, including mathematics, computer science, and probability.

## Example of using a combinations calculator

Let’s say you are a baker and you want to know how many different combinations of cupcakes you can make with 10 different flavors of cupcakes. You could use a combinations calculator to do this.

To do this, you would enter the following information into the calculator:

- Number of items in the set: 10
- Number of items in the combination: 3

The calculator would then display the following result:

```
Number of combinations: 120
```

Therefore, there are 120 different combinations of 3 cupcakes that you can make with 10 different flavors of cupcakes.

Combinations calculators can be used to calculate the number of combinations of any set of items, regardless of the size of the set. This makes them a valuable tool for a variety of applications.