Counting Outcomes

Combination

A selection of a set of objects, regardless of the selection order.

Permutation

A linear arrangement of a set of objects, with regard to the order of the arrangement.

Circular Permutation

A circular arrangement of a set of objects, with regard to the order of the arrangement. Unlike linear arrangements, circular permutations do not have a fixed starting point.

Rule 1

Suppose there is an experiment consisting of number of steps. And -th step having number of outcomes.

Rule 2

Total number of combinations of objects taken at a time is:

Rule 3

The number of permutations of objects taken at a time is:

Rule 4

The number of circular permutations of objects taken from a group of objects:

Total number of linear permutations are divided by , the number of objects being arranged, to eliminate equivalent rotations.

Alternatively, it can be expressed as the product of the number of combinations, , and the factorial of , which represents the number of ways to arrange the remaining objects after fixing one as a reference point.

Rule 5

Suppose a multi-set contains items of different types. There are number of type objects. Total number of distinct permutations:

Rule 6

Suppose a multi-set contains items of different types. There are number of type objects. Total number of distinct combinations (of all sizes):

This includes an empty set as well.