Year 12 Probability: Permutations

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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.

Work through the sections in order, then try the linked Year 12 skill quizzes after each section.

Core idea

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

Worked example

When it is used

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

Worked example 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.