How are permutations and combinations related to each other?

In combinatorics, the field of mathematics that deals with object combinations, counting, and arranging, permutations and combinations are closely related notions. While they both count the number of possible arrangements or selections of components from a set, the methods for doing so are different.

Permutations

Permutations refer to the arrangements of elements in a specific order. The order of arrangement matters in permutations. For example, the permutations of the set ${A, B, C}$ would include the following:

Formula

The formula for permutation is:

Here, $n$ is the total number of elements and $r$ is the number of elements taken at a time.

Let’s use the set example to apply the above formula. In the above set total number of elements are $3$, and the number of elements taken at a time is also $3$ because we want to find all permutations of the entire set.

So, $n = 3$ and $r = 3$, apply to the formula we have,

Combinations

Combinations refer to the selection of elements without considering the order. When using combinations, the sequence of selection is irrelevant. For example, the combinations of the set ${A, B, C}$ would include the following if we want to take only 2 elements at a time:

Formula

The formula we use for combination is:

Here, $n$ is the total number of elements and $r$ is the number of elements taken at a time.

Let’s use the set example to apply the above formula. In the above set, total number of elements are $3$, and the number of elements taken at a time is also $2$ because we want to find combinations of 2 elements from the set.

So, $n = 3$ and $r = 3$, apply to the formula we have,

Note: For permutations, we use the phrase arrangements, while for combinations, we use the word selections.

Relation

In combinations, a group is formed, and the arrangement order of the objects within the group is irrelevant, while permutation considers both the formation of a group and an arrangement in a certain sequence.

For example, although ab and ba are distinct permutations, they both refer to the same combination.

Now, let’s use the relation between permutations and combinations:

or we can also write the above formula as: