Combinations and permutations describe how many ways a number of objects taken from a group can be arranged. The number of objects in the group is written as *n*, and the number of objects to be arranged is represented by *r *(or *k*).

In a **combination**, the order of the selections does not matter because every available slot to be filled is the same. Examples of combinations include:

- choosing 3 people from a group of 12 to form a committee (220 possible committees)
- choosing 3 pizza toppings from 10 options (120 possible pizzas)

In a **permutation**, the order of the selection matters, meaning each available slot is different. Examples of permutations include:

- awarding gold, silver, and bronze medals in a race with 100 participants (970,200 possibilities)
- selecting a president, vice-president, secretary, and treasurer from a committee of 12 people (11,880 possibilities)

The formulas for both calculations are similar. The only difference—the *r*! in the denominator of a combination—accounts for redundant outcomes. Note that both permutations and combinations can be written in several different shortened notations.