What Is Permutation Without Repetition?
Permutation without repetition counts the number of ways to arrange r items selected from n distinct items, where each item can only appear once in any arrangement. This is the most common type of permutation encountered in mathematics and statistics.
Consider selecting 3 officers (president, vice-president, treasurer) from a club of 10 members. Once someone is chosen as president, they cannot also be vice-president. This constraint -- no repetition -- is what distinguishes this from permutations with repetition.
Formula
The formula can be understood intuitively: for the first position there are n choices, for the second n-1 (one item used), for the third n-2, and so on until r positions are filled.
Examples
| Scenario | n | r | P(n,r) |
|---|---|---|---|
| 3-digit lock (no repeat digits) | 10 | 3 | 720 |
| Podium finish from 8 runners | 8 | 3 | 336 |
| Seating 5 people in 5 chairs | 5 | 5 | 120 |
| Arranging all 26 letters | 26 | 26 | 26! |
With vs Without Repetition
- Without repetition: Each item used at most once. P(n,r) = n!/(n-r)!
- With repetition: Items can repeat. Result = n^r (n raised to power r).
- Without repetition always yields fewer arrangements when r > 1.
- When r = 1, both formulas give the same result: n.
Frequently Asked Questions
What happens when r equals n?
When r = n, you are arranging all items, giving n! permutations. For example, arranging 5 books on a shelf gives 5! = 120 different arrangements.
Can r be greater than n?
No. Without repetition, you cannot choose more items than are available. The formula requires r ≤ n. If you need r > n, use permutation with repetition instead.
What real-world problems use this?
Election results (ranking candidates), scheduling (ordering tasks), tournament brackets, license plates with non-repeating characters, and seating arrangements all use permutations without repetition.