N Choose R Order Does Not Matter

To calculate the number of possible combinations of choosing elements from a set of elements, where order does not matter, the combination formula is used.

For example, if we want to calculate the possible combinations of choosing two items from the set {♥︎,♦︎,♣︎,♠︎} we can set up the formula. Substitute into the equation which represents the size of the set and substitute into the formula which represents the number of items we are choosing.

Evaluate the subtraction.

Expand the factorial operator.

Simplify the fraction.

There are ways to select two items from the set {♥︎,♦︎,♣︎,♠︎}. This can be seen in the list of possible combinations shown below.

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

Example: 4 Choose 3

This example shows how to calculate the possible ways to select three items from the set of four represented as {♥︎,♦︎,♣︎,♠︎}.

  1. Start by setting up the combination formula.

  2. Substitute into the equation which represents the size of the set and substitute into the equation which represents the number of items we are choosing.

  3. Evaluate the subtraction in the denominator.

  4. Expand the factorial operator.

  5. Simplify the fraction

    There are possible ways to select three items from the set {♥︎,♦︎,♣︎,♠︎} shown in the list below.

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

Permutation vs. Combination

When selecting elements from a collection of elements, a distinction is made whether order does or does not matter. When order does not matter the combination formula is used.

When order does matter, the permutation formula is used. Note, the subscript and are used to denote permutations versus combinations.