Year 12 Probability: Permutations

Permutations appear in probability when outcomes are arrangements and the order matters. They are a counting tool, but they matter in probability because they help you measure the size of the sample space and the favourable outcomes.

Ask one question first: does order matter? If yes, you are in permutation territory.

Core idea

The number of ordered selections of $r$ objects from $n$ distinct objects is $P (n, r) = n! / (n - r)!$ .

Worked example

Problem: Five books A, B, C, D and E are arranged randomly on a shelf. What is the probability that A and B are next to each other?

Total arrangements = $5 \neq 120$ .
For favourable arrangements, treat A and B as one block.
Then the objects are: AB-block, C, D, E. These can be arranged in $4 \neq 24$ ways.
Inside the block, A and B can be arranged as AB or BA, so multiply by 2.
Favourable arrangements = $2 \times 4 \neq 48$ .
Probability = $48 / 120 = 2 / 5$ .

Answer: the probability that A and B are adjacent is

2 / 5

When it is used

Seating arrangements.
Digit arrangements.
Ordered selections from a group.
Counting probability outcomes where position changes the outcome.

Worked example 2

Problem: Four different runners line up randomly for a race. What is the probability that runner A finishes ahead of runner B?

Total orders of finish are $4 \neq 24$ .
By symmetry, A is equally likely to finish ahead of B as behind B.
That means exactly half of all arrangements have A ahead of B.
So the probability is $1 / 2$ .

Answer: the probability that A finishes ahead of B is

1 / 2

Common mistakes

Using combinations when the positions are actually different outcomes.
Forgetting to account for arrangements inside a block such as AB or BA.
Counting favourable outcomes correctly but using the wrong total sample space.

Quick check

Whenever a probability question uses the words arrange, order, line up, seat, or rank, stop and ask if the order affects the outcome. That single decision usually tells you whether permutations are the right tool.

Revision focus

Probability work becomes far more reliable when you define the object of the question before calculating. Is the problem about ordered outcomes, unordered selections, a conditional event, a discrete random variable, or a continuous density? That decision controls the method. In many exam questions, the major error is not arithmetic. It is using a correct formula for the wrong kind of probability situation.

A second useful habit is to interpret the result immediately after it is found. Decide whether the answer is a count, a probability, an average outcome, or a spread measure. In continuous settings, remember that probabilities come from areas over intervals, not from single points. In discrete settings, remember that the total probability must still add to one. Those interpretation checks are often the fastest way to catch an error before it reaches the final line.