Permutations

Factorial notation

For a positive integer $n$,

and $0!=1$. The number of ways to arrange $n$ distinct objects in a row is $n!$.

For example, $5!=120$ different orders are possible for five distinct books.

Arranging r objects chosen from n

The number of ordered selections of $r$ distinct objects from $n$ distinct objects is

This equals

the multiplication principle across $r$ positions.

Worked example

Selecting the same four people in different roles creates a different outcome.

Decide whether order matters

Ask: if the same chosen people or objects swap positions, is the outcome considered different? A podium finish, password, schedule, or set of assigned roles is ordered. A committee with equal membership roles is not.

Words such as “arrange,” “rank,” “seat,” and named offices usually signal permutations.

Common mistakes

Using a permutation when roles do not matter. A plain committee is a combination.

Using $n^r$ when repetition is forbidden. Available choices decrease.

Treating identical copies as distinct. Divide by their internal factorials.

Using $n!$ for a circular arrangement. Fix rotational symmetry first.

Counting a restricted block but forgetting its internal orders. Arrange both levels.

Quick self-check

• Does swapping selected objects create a new outcome?
• Are all objects distinct, or are some identical?
• Is selection with or without replacement?
• Are there adjacency, separation, or position restrictions?
• Are rotations or reflections considered equivalent?
• Does a multiplication-principle count confirm the formula?