Permutation Calculator
Calculate permutations P(n,r) — the number of ordered arrangements of r items chosen from n items.
Quick Answer
P(n,r) = n! / (n−r)!. For example, P(5,3) = 5! / 2! = 120 / 2 = 60 ordered arrangements.
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About This Tool
The Permutation Calculator computes P(n,r) — the number of ways to arrange r items from a set of n items where the order of arrangement matters. This differs from combinations, where order is irrelevant. Permutations are fundamental to probability, cryptography, and counting problems.
The Permutation Formula
P(n,r) = n! / (n−r)!. The numerator counts all possible orderings of n items. The denominator removes the orderings of the (n−r) items that were not selected. The result gives you only the arrangements of the r selected items.
Permutations vs. Combinations
The key difference: in permutations, ABC and CBA are different arrangements. In combinations, they count as the same group. P(n,r) is always greater than or equal to C(n,r), with P(n,r) = C(n,r) × r!. Use permutations when the position or order of the selection matters.
Real-World Applications
PIN codes, passwords, race finishing orders, phone numbers, and license plates all involve permutations. A 4-digit PIN from digits 0–9 has P(10,4) = 5,040 possibilities (without repetition). With repetition allowed, it becomes 10^4 = 10,000. Understanding which model applies is crucial for security analysis.
Permutations with Repetition
This calculator handles permutations without repetition. When items can repeat (like digits in a PIN), the formula is n^r instead. A 4-character password from 26 letters with repetition has 26^4 = 456,976 possibilities, compared to P(26,4) = 358,800 without repetition.
Frequently Asked Questions
What is a permutation?
How do permutations differ from combinations?
What is P(n,n)?
How do I calculate permutations with repetition?
Why is P(n,r) always larger than C(n,r)?
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