nPr Permutation Calculator P(n,r)

P(n, r) Result

What is a Permutation?

A permutation is an ordered arrangement of items. P(n,r) gives the number of ways to arrange r items chosen from n items, where the order matters.

P(n, r) = n! / (n − r)!
P(10, 3) = 10! / 7! = 10 × 9 × 8 = 720

When Order Matters: Use Permutations

  • Number of ways to arrange 3 people in 3 seats from 10 people
  • First, second, third place in a race from 8 runners
  • 4-digit PIN codes from 10 digits (with repetition not allowed)

Permutation vs Combination — Key Difference

FeaturePermutation P(n,r)Combination C(n,r)
Order matters?YESNO
ABC = BCA?Different arrangementsSame selection
Formulan!/(n−r)!n!/[r!(n−r)!]
P(5,2)20 arrangementsC(5,2)=10 selections
Use whenRankings, codes, sequencesTeams, groups, sets

Worked Examples

Example 1 — Race Podium

8 runners, find P(Gold, Silver, Bronze):
P(8,3) = 8×7×6 = 336 ways

Example 2 — Password

4-digit PIN from digits 0−9 (no repeat):
P(10,4) = 10×9×8×7 = 5,040 combinations
(With repeats allowed: 10âī = 10,000)

Example 3 — Arrangement

Arrange all 5 letters A,B,C,D,E:
P(5,5) = 5! = 120 arrangements

Real-World Applications

  • Sports: Top-3 finishers in a race from 10 athletes = P(10,3) = 720
  • Security: Combination lock (misnomer — actually a permutation!) with 3 numbers from 40: P(40,3)
  • Scheduling: Ordering of tasks or appointments
  • Cryptography: Counting possible arrangements in cipher systems

Permutations with Repetition

When items can be repeated (like digits in a PIN), the formula changes:

With repetition: nĘģ arrangements
4-digit PIN from 0–9 WITH repeats: 10âī = 10,000
4-digit PIN from 0–9 WITHOUT repeats: P(10,4) = 5,040

Circular Permutations

When arranging n items in a circle (like people around a round table), we fix one position as reference, giving (n−1)! arrangements instead of n!:

Circular permutations of n items = (n-1)!
6 people at a round table = 5! = 120 ways
(vs 6! = 720 for a straight line)

Permutation Quick Reference

nP(n,1)P(n,2)P(n,3)P(n,n)=n!
33666
44122424
552060120
6630120720
1010907203,628,800

Frequently Asked Questions

What is P(5,2)? ▾
P(5,2) = 5!/(5−2)! = 5!/3! = 5×4 = 20. There are 20 ordered ways to choose 2 items from 5.
What is the difference between permutation and combination? ▾
In permutations, order matters (ABC ≠ BCA). In combinations, order does not matter (ABC = BCA). Use permutations for rankings, arrangements; combinations for selections, groups.
What is P(n,n)? ▾
P(n,n) = n!/0! = n!/1 = n!. It gives the number of ways to arrange all n items, which equals n factorial.