# Combinations Calculator (nCr)

Enter the number of elements (n) and the sample number (r).

*n* choose *r*, \(C(n,r)={n\choose r}\).

**Answer:**

**Step-by-step solution:**

Use this calculator to find the result of a combination. Enter the total number of items and the number of items chosen.

## How to use the combination calculator?

**Step 1:** Enter the total number of elements in the first input box. This is the value of *n*.

**Step 2:** Enter the number of elements chosen from the set in the second input box. This is the value of *r*.

**Step 3:** Click “Calculate” to get the result of the combination.

**Step 4:** The answer will be displayed on the right and the step-by-step solution will be displayed at the bottom.

## What types of numbers can I enter into the calculator?

The values of *n* and *r* must be positive and integers. Since we are talking about sets of elements, we cannot have negative or fractional numbers to calculate combinations.

Also, the value of *r* must be less than or equal to the value of *n*. The value of *n* is the total number of elements and the value of *r* is the number of elements chosen, so *r* cannot be greater than *n*.

Therefore, taking into account these two conditions, the entered numbers must follow the following:

\(n\geq r\geq 0\)

## What are combinations?

A combination is a mathematical technique that determines the number of possible arrangements of elements, where the order does not matter. In combinations, we can select the elements in any order.

You can learn more about combinations by visiting our main article.

## How to find combinations?

Combinations can be found using the combinations’ formula. This formula determines the number of possible ways to select only a few objects from a set without repetition:

\( C(n,~r)={n \choose r}=\frac{n!}{r!(n-r)!}\)

where *n* is the total number of elements in the set and *r* is the number of chosen objects.

For example, if we have a set of 8 elements and we choose 6, we can calculate the number of combinations as follows:

\( C( n,~r)=\frac{n!}{r!(n-r)!}\)

\( C( 8,~6)=\frac{8!}{6!(8-6)!}\)

\( =\frac{8!}{6!(2)!}\)

Now, we can rewrite to 8! like 8×7×6!, to then simplify:

\( C( 8,~6)=\frac{8\times 7 \times 6!}{6!(2)!}\)

\( =\frac{8\times 7 }{2!}\)

\( =28\)

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