**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

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

Here are some 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.

## Use Cases

Combinations calculators can be used in various fields such as:

- Mathematics: Combinations calculators are widely used in mathematics for solving problems related to combinatorics.
- Computer Science: Combinations calculators are used in computer science for solving problems related to algorithms and data structures.
- Probability: Combinations calculators are used in probability theory for calculating probabilities.

**References**

Here are some references related to combinations:

- 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