Permutation & Combination Calculator

Calculate P&C problems

Permutations are arrangements where order matters. Formula: P(n,r) = n! / (n-r)!
Combinations are selections where order doesn't matter. Formula: C(n,r) = n! / (r!(n-r)!)
Factorial of n (n!) is the product of all positive integers from 1 to n. Example: 5! = 5×4×3×2×1 = 120
Quick Reference
Key Differences
  • Permutations: Order matters (ABC ≠ BAC)
  • Combinations: Order doesn't matter (ABC = BAC)
  • Example: Choosing team captain vs. choosing team members
Common Factorials
  • 0! = 1
  • 3! = 6
  • 5! = 120
  • 10! = 3,628,800

How the Permutation & Combination Calculator Works

Our permutation and combination calculator provides three calculation modes. The Permutations tab calculates the number of ordered arrangements of r items from a set of n, with an option for repetition. The Combinations tab calculates unordered selections of r items from n. The Factorial tab computes n! for any non-negative integer up to 170.

Each calculation displays not just the final answer but also the step-by-step formula expansion, helping students understand the underlying mathematics. The calculator handles large numbers accurately and provides clear explanations for every result.

Permutation & Combination Formulas

Permutation: P(n,r) = n! / (n−r)!
Combination: C(n,r) = n! / [r!(n−r)!]

In the permutation formula, we divide n! by (n−r)! to count only the arrangements of r items from the total n. Since order matters, different sequences of the same items are counted separately. With repetition allowed, the formula simplifies to n^r because each of the r positions can be filled by any of the n items independently.

For combinations, we further divide by r! to eliminate the duplicate counts caused by different orderings of the same selection. This is why C(n,r) is always less than or equal to P(n,r). The relationship is: C(n,r) = P(n,r) / r!. These formulas are foundational in probability theory, statistics, and discrete mathematics.

Frequently Asked Questions

The key difference is whether order matters. A permutation counts arrangements where order is important (e.g., passwords, rankings, seating arrangements), while a combination counts selections where order does not matter (e.g., choosing team members, lottery numbers). For example, choosing 3 people from 10 for president, VP, and secretary is a permutation (720 ways), but choosing any 3 people from 10 for a committee is a combination (120 ways).

A factorial, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials are fundamental to permutation and combination formulas. They grow extremely fast — 10! = 3,628,800 and 20! = 2,432,902,008,176,640,000. Factorials represent the total number of ways to arrange n distinct objects.

Use permutations when the order or arrangement of items matters — such as arranging books on a shelf, creating passwords, assigning ranked positions, or determining race finishing orders. Use combinations when you are simply selecting items and order does not matter — such as choosing lottery numbers, picking team members, selecting menu items, or forming committees. Ask yourself: does rearranging the same items create a different outcome? If yes, use permutations; if no, use combinations.

The total number of possible combinations from N items (choosing any number of them) is 2^N. This includes choosing 0 items, 1 item, 2 items, and so on up to all N items. For specific selections of r items from n, use C(n,r) = n! / [r!(n-r)!]. For example, from 5 items you can make 2^5 = 32 total combinations, while choosing exactly 3 from 5 gives C(5,3) = 10 combinations.

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