Permutations
Number of ordered arrangements of r items from n.
Formula
Variables
Example
Ways to arrange 2 of 4 people in order: P(4,2) = 4!/2! = 12.
How it works
This formula counts how many ways you can choose and arrange r distinct items from a larger set of n when order matters. The first position has n choices, the second has n−1 choices, and this continues until r positions are filled, giving n × (n−1) × ... × (n−r+1), which is exactly n!/(n−r)!. It works because once an item is used, it cannot be repeated in the arrangement.
When to use it
Use this when the order of selection matters, such as assigning first, second, and third place in a competition, choosing a password without repetition, or arranging people in seats. If order does not matter, use combinations instead.
Worked example, step by step
How many ways can 3 books be arranged from a shelf of 7 different books? P(7,3) = 7! / (7−3)! P(7,3) = 7! / 4! P(7,3) = (7 × 6 × 5 × 4!) / 4! P(7,3) = 7 × 6 × 5 P(7,3) = 210 So, there are 210 ordered arrangements.
Common mistakes to avoid
- Using the formula when order does not matter
- Forgetting that n and r must be whole numbers with r ≤ n
- Computing n! and (n−r)! separately instead of simplifying first
Did You Know?
Permutations count ordered arrangements — there are 8×10⁵⁶ ways to shuffle a deck of 52 cards.
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