Permutation Calculator (nPr) — Free Arrangement Calculator | AllInOneTools
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P(n,r) Combinatorics

Permutation Calculator

Calculate P(n,r) — the number of ordered arrangements of r items chosen from n. See step-by-step solution, multiplication chain, and comparison with combinations.

P (
n (total)
r (arrange)
) =
720
📑 Step-by-Step Solution
📈 Permutation vs Combination

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Permutations: Counting Ordered Arrangements

A permutation counts the number of ways to arrange r items selected from a set of n distinct items where order matters. Arranging items as {A, B, C} is different from {C, B, A}. This is the key difference from combinations, where order is irrelevant. Permutations are used whenever sequence, rank, or position matters: assigning medals (gold, silver, bronze), creating passwords, scheduling tasks, seating arrangements, license plates, and phone numbers. The formula P(n,r) = n!/(n-r)! can also be understood as a descending product: n × (n-1) × (n-2) × ... for r terms.

Permutation Formulas

Standard Permutation (without repetition):
  P(n,r) = n! / (n-r)!
  P(n,r) = n x (n-1) x (n-2) x ... x (n-r+1)
  (r terms in descending order)

Full Permutation (all items):
  P(n,n) = n! (arrange all n items)

Permutation with Repetition:
  n^r (each of r positions has n choices)
  Example: 4-digit PIN = 10^4 = 10,000

Multiset Permutation:
  n! / (n1! x n2! x ... x nk!)
  For items with duplicates

Relationship to Combinations:
  P(n,r) = C(n,r) x r!
  C(n,r) = P(n,r) / r!

Real-World Applications

In security, permutations determine password strength: a 4-digit PIN from digits 0-9 has P(10,4) = 5,040 arrangements without repetition, or 104 = 10,000 with repetition. Sports tournaments use permutations for rankings: awarding gold, silver, and bronze from 8 athletes gives P(8,3) = 336 possible outcomes. License plates with 3 letters and 4 digits use permutation counting: 263 × 104 = 175,760,000. In scheduling, arranging 5 tasks in order gives P(5,5) = 120 possible schedules. Cryptography relies on the enormous size of permutation spaces to ensure security.

Permutation vs. Combination Quick Guide
Ask: "Does the order of selection matter?" If YES, use permutation P(n,r). If NO, use combination C(n,r). Examples: Choosing a committee of 3 from 10 = C(10,3) = 120 (order does not matter). Choosing president, VP, treasurer from 10 = P(10,3) = 720 (positions are distinct). P(n,r) is always greater than or equal to C(n,r) because P(n,r) = C(n,r) x r!.

Frequently Asked Questions

What is a permutation? ▾
A permutation is an ordered arrangement. P(n,r) counts the ways to arrange r items from n where order matters. P(n,r) = n!/(n-r)!. Example: P(5,2) = 20 ordered pairs from 5 items.
What is the difference between permutation and combination? ▾
Permutations: order matters (AB differs from BA). Combinations: order irrelevant (AB = BA). P(n,r) = C(n,r) x r!. Permutations always give equal or larger numbers than combinations.
What is permutation with repetition? ▾
When items can be reused, each of r positions has n choices, giving n^r total arrangements. Example: 3-letter codes from 26 letters with repetition = 26^3 = 17,576.
How do I calculate P(n,r) without factorials? ▾
Multiply r descending numbers starting from n: P(n,r) = n x (n-1) x (n-2) x ... for r terms. Example: P(8,3) = 8 x 7 x 6 = 336. No need to compute full factorials.
Why does P(n,0) = 1? ▾
There is exactly one way to arrange zero items: do nothing (the empty arrangement). Mathematically, P(n,0) = n!/n! = 1 for any n.
What is a multiset permutation? ▾
When some items are identical, divide by the factorial of each group's count: n!/(n1! x n2! x ...). Example: arrangements of MISSISSIPPI = 11!/(4!4!2!) = 34,650.