Permutations

Use a permutation when the selected objects' order, roles, or positions matter.

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.

Repeated identical objects

If $n$ objects include identical groups of sizes $n_1,n_2,\ldots$, the number of distinguishable arrangements is

For the letters of LEVEL, there are $5!/(2!2!)=30$ arrangements because the two Ls and two Es are indistinguishable.

Objects kept together

Treat a required group as one block, arrange the block with the other objects, then arrange members within the block.

If two particular people must sit together among six people, treat them as one unit: $5!$ block arrangements times $2!$ internal orders.

Objects kept apart

Count all arrangements and subtract those with the objects together, or arrange other objects first and place separated objects in the gaps.

Choose the method that avoids overlap and makes restrictions explicit.

Circular permutations

For $n$ distinct objects around a circle where rotations are considered the same, fix one object and arrange the remaining $n-1$:

If reflections are also considered identical, as in some necklace problems, another adjustment may be necessary. Clarify the equivalence rule.

Permutations with repetition allowed

If each of $r$ ordered positions can use any of $n$ symbols and repetition is allowed, there are

sequences. This is not ${}_nP_r$, which assumes selection without replacement.

Probability applications

Count an ordered sample space and favourable ordered outcomes consistently. For a random arrangement, every permutation may be equally likely, but derived categories may not be.

State what constitutes one outcome before forming a probability ratio.

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?

Related topics

• Fundamental Counting Principle
• Combinations
• Probability
• Factorials