Permutation Calculator

About the Permutation Calculator

The Permutation Calculator computes P(n, r), the number of ordered arrangements of r items chosen from n distinct items. Unlike combinations, permutations care about the order of selection.

Permutations answer questions like "How many 3-letter codes can be made from 26 letters without repeats?" or "In how many ways can the top 3 finishers be arranged in a race of 10 runners?"

This calculator computes P(n, r) = n! / (n−r)! and compares it with the combination count so you can see how order multiplies the possibilities.

When This Page Helps

Permutation calculations involve large factorials. This calculator computes the result efficiently and compares with combinations to clarify the effect of ordering.

How to Use the Inputs

1. Enter n (total items).
2. Enter r (items to arrange).
3. View P(n, r) = n! / (n−r)!.
4. Compare with C(n, r) to see how order matters.
5. r must be ≤ n.

Example Calculation

Tips & Best Practices

• P(n, n) = n! (all items arranged in order).
• P(n, 1) = n (simply choosing one item).
• P(n, r) = C(n, r) × r! — each combination has r! orderings.
• Lock combinations are actually permutations (order matters).
• Phone PINs are permutations with repetition: 10^4 = 10,000.
• Use permutations when the arrangement or sequence matters.

Permutations in the Real World

Race finishes, password possibilities, seating arrangements, and tournament brackets all involve permutations. Any situation where the order of selection matters requires permutation counting.

Permutations of Multisets

When some items are identical, the formula becomes n! / (n₁! × n₂! × ...). The word MISSISSIPPI has 11! / (4! × 4! × 2!) = 34,650 distinct arrangements.

Circular Permutations

For arrangements in a circle (like seating at a round table), the count is (n−1)! because rotations of the same arrangement are considered identical.

Mastering this concept provides a strong foundation for advanced coursework in mathematics, statistics, and related quantitative disciplines.

Frequently Asked Questions

• What is a permutation?

  A permutation is an ordered arrangement of items. ABC, ACB, BAC are three different permutations of the same three letters.

• Permutations vs combinations?

  Permutations consider order (ABC ≠ CBA). Combinations do not (ABC = CBA). P(n,r) is always ≥ C(n,r), with equality only when r ≤ 1.

• What about permutations with repetition?

  When items can repeat, the count is nʳ (n to the power of r). A 4-digit PIN from digits 0-9 has 10⁴ = 10,000 possibilities.

• What is a derangement?

  A derangement is a permutation where no element appears in its original position. The number of derangements of n items is approximately n!/e.

• How are permutations used in cryptography?

  Encryption algorithms use permutations to shuffle data. The security of many ciphers depends on the infeasibility of testing all possible permutations.

• Can r be 0?

  Yes, P(n, 0) = 1. There is exactly one way to arrange zero items: the empty arrangement.