Calculates the number of combinations of p elements from a set of n elements : `([n], [p]) \text{ }(n >= p)`

This online tool calculates the number of combinations of p-elements from a set of n elements. In combinatorics, an important element that defines a combination is that the order of elements does not matter.

What is a Combination ?

A combination of p elements from n is an unordered selection of p elements from n elements.

- {2,3,1 is a combination of three elements out of {1,2,3,4} but {2,1,3} represents the same combination because the order does not matter.
- {c, a} is a combination of two elements from {a, b, c}. Similarly, {a, c} represents the same combination.

Combination calculation formula

Given a set of n different objects then, the number of combinations of p objects from n is equal to,

`([n], [p]) = frac {n!}{p! * (n-p)!}`

Example: E is a set of three elements 1, 2, 3. The number of combinations of two elements from E is equal to 3! / (2! . (3-2)!) = 3. These combinations are :
1, 2,
1, 3,
2, 3

How to identify a combination ?

There are 2 criteria in recognizing a combination :
- Order does not matters
- Only a subset of the whole set is concerned

Example: How many possible draws are there in a 5/49 lottery ?
We need to calculate a combination because the draw order is not important and only a subset of numbers is concerned (5 of 49).
The number of possible draws is 49! / (5! (49-5)!) = 1 906 884.

Comparison of enumeration methods

Method Elements concerned Order matters ? Formula
Permutation All elements (n) Yes `n!`
Arrangement Subset of p elements among n Yes `frac {n!}{(n-p)!}`
Combination Subset of p elements among n No `frac {n!}{p!* (n-p)!}`

See also

Binomial coefficient