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P Permutation Calculator — P(n, r)

Calculate the number of ordered arrangements (permutations) of r items chosen from n items. P(n, r) = n! / (n − r)!. Enter n and r to get the result, step-by-step working, and all possible arrangements for small values.

Number of items in the set (0–200)

Number of items to select and arrange

Permutation Formula

P(n, r) = n! / (n − r)!
= n × (n−1) × (n−2) × ... × (n−r+1)

A permutation is an arrangement where order matters. Choosing {A, B, C} is different from {C, B, A}. Use combinations when order does not matter.

Permutation Examples

P(n, r) Calculation Result
P(5, 3) 5 × 4 × 3 60
P(4, 2) 4 × 3 12
P(6, 6) 6! 720
P(10,3) 10 × 9 × 8 720
P(52,2) 52 × 51 2,652

How to Calculate P(5, 3)

  1. 1
    Write the formula
    P(5, 3) = 5! / (5−3)! = 5! / 2!
  2. 2
    Expand the numerator
    5! = 5 × 4 × 3 × 2 × 1 = 120
  3. 3
    Expand the denominator
    2! = 2 × 1 = 2
  4. 4
    Divide
    120 / 2 = 60. There are 60 ordered arrangements of 3 items from 5.

Frequently Asked Questions

A permutation is an ordered arrangement of items selected from a set. The order of selection matters — {A, B} and {B, A} are counted as two different permutations. P(n, r) counts all possible ordered arrangements of r items chosen from a pool of n distinct items.

In a permutation, order matters. In a combination, it does not. P(5, 3) = 60 but C(5, 3) = 10. Every combination of r items corresponds to r! permutations, so P(n, r) = C(n, r) × r!.

Order matters when the arrangement itself is significant: rankings (1st, 2nd, 3rd place), passwords, scheduling tasks in sequence, or assigning distinct roles to people. If you just need to pick a group without roles, use combinations.

Medal standings in a race (gold/silver/bronze), arranging letters in a word, assigning unique ID codes, scheduling jobs on a machine, planning a playlist order, and seating people in numbered chairs all involve permutations.

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