Key takeaway: A permutation is an ordered arrangement of r items taken from n distinct items, counted by P(n, r) = n!/(n − r)!, where changing the order gives a new result.
Grade 12 – PrecalculusS-CP.B.9Probability and Counting
Definition of Permutation
A permutation counts arrangements in which swapping two positions creates a new outcome. A podium is a classic case: the same three athletes can stand in several orders, and each order is a different result.
To count them, fill the positions one at a time. The first position can be filled in \(n\) ways, the second in \(n-1\) ways because one item is already used, and so on for \(r\) positions. Multiplying these shrinking choices gives the total. When every item is placed, the count reduces to the factorial \(n!\).
Formula and notation
The number of ordered selections of \(r\) items from \(n\) distinct items is
\[P(n,r)=\frac{n!}{(n-r)!}=n(n-1)(n-2)\cdots(n-r+1).\]
Other labels in use are \(_nP_r\) and \(P_r^n\). Both mean the same count, and \(P(n,n)=n!\).
Key facts
- Use a permutation when the order of selection changes the outcome.
- Prizes, passwords, seating charts and race finishes are typical permutation situations.
- P(n, 1) = n and P(n, n) = n!, the two simplest cases.
- Dividing a permutation count by r! gives the count of unordered groups.
Permutation example
Eight sprinters line up for a relay final. How many different top-three rankings could the scoreboard show?
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Order matters, so count permutations of 3 from 8: \(P(8,3)=\frac{8!}{5!}\).
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Cancel the shared tail: \(8\cdot 7\cdot 6\).
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Multiply: \(8\cdot 7=56\), and \(56\cdot 6=336\).
The scoreboard could show 336 different rankings.
Common mistake
Watch out
Priyanka needs the number of ways to pick a 3-person relay squad from 8 runners and also answers 336.
Why it is wrong
A squad has no ranking, so a permutation overcounts: each squad appears \(3!=6\) times , once per order. The count of squads is \(\frac{336}{6}=56\). Decide first whether order matters.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Probability and Counting