# Combinations (nCr) Calculator

LAST UPDATE: September 24th, 2020

## Definition – What are combinations (nCr)?

Combinations (nCr) are the number of combinations of numbers that can be put together where the order that they are selected doesn’t matter and numbers are not repeated.
For example, a set of 3 numbers (1, 2, and 3) with combinations of 2 would have 3 different combinations:
1 2
1 3
2 3
A set a 4 numbers (1, 2, 3, and 4) with combinations of 2 would have 6 different combinations:
1 2
1 3
1 4
2 3
2 4
3 4
A set of 4 numbers (1, 2, 3, and 4) with combinations of 3 would have 4 different combinations:
1 2 3
1 2 4
2 3 4
1 3 4

## Formula – How are combinations calculated?

Combinations are calculated using the formula:

numbers! ÷ (number of items! x (numbers – number of items)!)

### Example

There are 20 different numbers, and each combination has 3 numbers. How many different combinations are there?

Combinations = 20! ÷ (3! x (20-3)!)

Combinations = 2432902008176640000 ÷ (6 x 17!)

Combinations = 2432902008176640000 ÷ (6 x 355687428096000)

Combinations = 2432902008176640000 ÷ 2134124568576000

Combinations = 1140