Permutations and combinations both count the number of ways to select a group of items from a larger set, and the formulas look almost identical — but they answer genuinely different questions depending on one crucial factor: does the order of selection matter?
If arranging three people in first, second, and third place counts ABC differently from BCA, you're dealing with permutations — order matters. If you're just selecting a 3-person committee from a group, where ABC and BCA represent the exact same committee, you're dealing with combinations — order doesn't matter. This single distinction determines which formula applies.
The Two Formulas
Permutations (order matters): nPr = n! ÷ (n − r)!
Combinations (order doesn't matter): nCr = n! ÷ [r! × (n − r)!]
Notice combinations is just permutations divided by r! (the number of ways to arrange the r selected items among themselves) — which is exactly why nPr is always greater than or equal to nCr for the same n and r: combinations removes the "different orderings of the same group" duplicates that permutations counts separately.
A Worked Example
Choosing 3 people from a group of 10, ranked for 1st, 2nd, and 3rd place (order matters): P(10,3) = 10! ÷ 7! = 10 × 9 × 8 = 720 possible ordered outcomes.
Choosing the same 3 people from a group of 10, just as an unranked committee (order doesn't matter): C(10,3) = 10! ÷ (3! × 7!) = 720 ÷ 6 = 120 possible unordered groups.
Both used the same 10 people and the same group size of 3 — the only difference is whether order was counted, and it changes the result by a factor of 6 (which is 3!, the number of ways to arrange 3 items).
Common Mistakes to Avoid
- Using the wrong formula for the situation: always ask first whether order matters in the specific scenario before picking permutations or combinations.
- Forgetting combinations will always produce a smaller number than permutations for the same n and r: if your combination calculation somehow produces a larger number than the equivalent permutation, something's wrong.
- Applying these formulas when repetition is allowed: both standard formulas assume each item can only be selected once — scenarios allowing repeated selection (like a 4-digit PIN) require different formulas entirely.
- Not simplifying factorial expressions before calculating: n! ÷ (n−r)! simplifies to a much shorter multiplication than computing both full factorials separately and dividing.
Bottom Line
The order-matters question is the entire key to choosing between permutations and combinations. Use a Permutation Calculator for ordered arrangements, and check the combination formula whenever the group itself — not the arrangement — is what counts.