N Choose R Combination Notation

N Choose R CombinationNotation

The number of possible ways to choose combinations from total items is denoted using two parentheses with the variables and . The subscript is included to indicate that the expression represents combinations where order does not matter as opposed to permutations where order does matter. Typically, the notation is used in an expression like this:

In plain language, this represents the total number of combinations from choosing items from a collection of items. See permutations for where order does matter.


Combination | Formula

The combination formula describes the possible combinations of r elements out of a group of n elements where order does not matter.


To calculate four choose two where order does not matter, you can use the formula for combinations. For example if we want to calculate the possible combinations of choosing two items from the set {♥︎,♦︎,♣︎,♠︎} we can set up the formula:

First we substitute four in for the variable , representing the size of the set. Then we substitute two for the variable , representing the number of items we are choosing.

To double check our work, we can display the six possible permutations of these two items chosen from the set. Note, if order does matter, you can use the formula for permuations.

  1. { ♥︎, ♠︎ }
  2. { ♦︎, ♠︎ }
  3. { ♥︎, ♦︎ }
  4. { ♣︎, ♠︎ }
  5. { ♥︎, ♣︎ }
  6. { ♣︎, ♦︎ }