# Combinations Calculator (nCr)

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

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

#### 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.

## 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:

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

$$=\frac{8\times 7 }{2!}$$

$$=28$$

## Related calculators:

You can explore other calculators here.