How to count permutations

A permutation counts the number of ways to arrange items when the order matters, that is, when swapping two items gives a different arrangement. In the Form 5 Permutation and Combination chapter of Add Math, this is the tool for questions such as 'in how many ways can 3 books be arranged in a row from 5 different books?'

or 'how many 3-digit codes can be formed?'. The key formula is ${}^{n}P_{r}=\dfrac{n!}{(n-r)!}$, where $n$ is the number of distinct items available and $r$ is the number of positions to fill.

It answers counting problems where each ordering is treated as a separate outcome, and it feeds directly into probability questions later on.

Reach for a permutation whenever a question is about arranging or ordering objects, or filling positions in a line, a row, or a sequence, look for words like 'arrange', 'in a row', 'order', 'line up', or 'code'. The deciding test is simple: ask whether changing the order gives a different result.

If AB is different from BA, order matters and you use a permutation; if AB is the same as BA (a plain selection), you use a combination instead. Seat arrangements, forming numbers from digits, and lining up people or books are all permutation problems, so read the question for whether position or order is important.

1. 1

   Check that order matters

   Confirm that a different order counts as a different arrangement, if so, it is a permutation.

2. 2

   Identify n and r

   Let $n$ be the number of distinct items available and $r$ the number of positions to fill.

3. 3

   Substitute into the formula

   Write ${}^{n}P_{r}=\dfrac{n!}{(n-r)!}$ with your values of $n$ and $r$.

4. 4

   Cancel the factorials

   Expand only as far as needed and cancel the common $(n-r)!$.

5. 5

   Compute the value

   Multiply the remaining factors to get the final count.

• Using a combination ${}^{n}C_{r}$ when the order actually matters, or a permutation when it does not.
• Swapping $n$ and $r$ in the formula, e.g. writing $\dfrac{r!}{(n-r)!}$.
• Forgetting that repeated identical items reduce the count, arrangements of a word with repeated letters need division by the factorials of the repeats.
• Misreading whether all $n$ items are used or only $r$ of them.
• Arithmetic slips when expanding factorials, cancel the common factorial first to keep the numbers small.

The step that trips students up is the very first one: deciding whether order matters. Choose wrongly and the whole answer uses the wrong formula.

In a one-to-one lesson our teachers give you a quick, reliable test,'does swapping two items make a new arrangement?', and practise it on mixed questions until the choice between ${}^{n}P_{r}$ and ${}^{n}C_{r}$ is instant. We also show how cancelling the common factorial keeps the arithmetic clean.

Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English. To see how we teach permutations, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.