Overview
Permutations and combinations both count ways to select items from a group, but they differ on whether the order of selection matters. Permutations count ordered arrangements (like ranking contestants 1st, 2nd, 3rd). Combinations count unordered groups (like choosing 3 team members from 10).
The Permutation Calculator computes the number of ways to arrange items where order matters. The Combination Calculator computes the number of ways to choose items where order does not matter.
Key Differences
**Order:** Permutations are order-sensitive (ABC differs from BAC). Combinations are order-insensitive (ABC = BAC).
**Formula:** Permutations: P(n,r) = n!/(n-r)!. Combinations: C(n,r) = n!/[r!(n-r)!]. Combinations divide by r! because order does not matter.
**Result size:** Permutations always produce a larger number than combinations for the same n and r (unless r = 0 or r = 1).
**Examples:** Permutation: How many ways can 3 runners finish a race from 10 entrants? Combination: How many ways can you choose 3 books from 10?
**Key question to ask:** "Does rearranging the same items create a different outcome?" If yes, use permutations. If no, use combinations.
When to Use the Permutation Calculator
- You are counting arrangements where position or rank matters - You need to know how many ways to assign 1st, 2nd, 3rd place from a group - You are calculating possible password combinations (where character order matters) - You are determining the number of ways to seat people in specific chairs - You need to count possible orderings of a sequence
When to Use the Combination Calculator
- You are choosing a subset where the order of selection does not matter - You want to know how many lottery ticket combinations exist - You are selecting committee members, team rosters, or ingredients from a list - You need to count handshakes (each pair is one handshake regardless of who reaches first) - You are solving probability problems involving "choose" scenarios
Frequently Asked Questions
Q: How do I know if order matters? A: Ask yourself: "Would swapping two selected items create a different result?" If choosing pizza toppings, swapping pepperoni and mushrooms gives the same pizza (combination). If assigning first and second place, swapping them changes the result (permutation).
Q: Why are permutations always larger? A: Because every combination can be rearranged into multiple permutations. For example, the combination {A,B,C} can be arranged in 6 permutations (ABC, ACB, BAC, BCA, CAB, CBA).
Q: What does the "!" mean in the formulas? A: The exclamation mark means factorial -- multiplying a number by every positive integer below it. 5! = 5 x 4 x 3 x 2 x 1 = 120.
Q: Can either calculator handle large numbers? A: Yes. Both tools compute results for large n and r values, though very large factorials produce extremely large numbers.
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