Permutation & Combination Calculator

⚡ Quick Tips for Picking the Right Formula

Features

Order Matters, or It Doesn't — the One Question That Decides Everything

Every permutation-versus-combination confusion collapses once you ask a single question: does rearranging the same items produce a genuinely different result? Three runners crossing the finish line as gold, silver, bronze is a permutation — swap who's first and third and the outcome is clearly different. Three people chosen for a committee from a larger group is a combination — the same three people, regardless of the order they were picked in, form the identical committee. Permutations of any given n and r are always exactly r! times larger than the matching combinations, because every unordered group of r items can itself be arranged in r! different orders — which is precisely the relationship this calculator shows worked out with real numbers, not just stated as an abstract rule. Getting this one question backwards is the single most common combinatorics mistake — treating a selection problem as if order mattered inflates the count by a factor of r!, and the reverse mistake shrinks a genuinely order-sensitive count down to a fraction of the real answer.

Why "With Repetition" Needs a Completely Different Formula

A standard permutation or combination assumes each item gets used at most once — but plenty of real counting problems don't work that way. A 4-digit PIN allows 1-1-1-1 as a valid code, so counting every possible PIN uses n^r (10,000 possible codes from 10 digits), not the standard permutation formula, which would wrongly disallow repeated digits entirely. Choosing 3 scoops from 5 ice cream flavors, allowed to repeat a flavor, needs yet another formula — C(n+r−1, r), often called the stars-and-bars method, which handles the fact that "chocolate, chocolate, vanilla" and "chocolate, vanilla, chocolate" describe the same order, the same 3 scoops. Getting repetition wrong in either direction — allowing it when a problem forbids it, or forbidding it when a problem allows it — is one of the most common combinatorics mistakes, which is exactly why this calculator makes it a single explicit switch rather than a separate hidden assumption.

When Some of Your Items Are Identical

Plain factorial assumes every item you're arranging is distinguishable from every other — but a word's own letters are rarely unique. MISSISSIPPI has 11 letters, and if each were genuinely distinct there would be 11! arrangements, but since the four I's are identical to each other and the four S's are identical to each other, swapping two I's produces no visible change at all. Multiset Permutations divides those indistinguishable swaps back out — 11! divided by 4! for the I's and another 4! for the S's — landing on exactly 34,650 truly distinct arrangements. The same logic applies well beyond words: scheduling identical shifts, arranging a rack of same-colored tiles, or any situation where "identical" items make some rearrangements invisible.

Circular Arrangements — Why a Round Table Isn't Just n!

Seat 5 people in a straight line and there are 5! = 120 distinct arrangements, because each of the 5 positions is genuinely different from the others — first chair, last chair, everything between. A round table breaks that assumption: rotate every single person one seat to the left, and the seating chart looks completely unchanged — the same person still sits to your left, the same person still sits to your right, even though everyone technically moved. Counting rotations as the "same" arrangement means fixing one person's seat as a reference point and only counting how the remaining n−1 people can be arranged relative to them, which is exactly why the formula becomes (n−1)! instead of n! — for 5 people, that's 24 distinct seatings, a fifth of the straight-line count. This calculator surfaces it as a bonus result any time n is entered, since it's a genuinely common follow-up question the moment "arranging items" turns into "arranging items around something circular" — a table, a keychain, a rotating schedule.

Pascal's Triangle Is a Combinatorics Cheat Sheet in Disguise

Every entry in the triangle is a combination in costume — row n, position k is exactly C(n,k), the same number nCr mode would compute from those two inputs directly. That single fact explains both visual patterns the triangle shows off: each row reads identically forwards and backwards because C(n,k) always equals C(n,n−k), and each row sums to exactly 2ⁿ because that sum is literally counting every possible subset — including the empty one and the full set — of an n-item collection. Rather than a separate curiosity bolted onto a combinations calculator, it's the same underlying math laid out as a shape instead of a formula.

Where People Actually Use Permutations and Combinations

Lottery and raffle organizers use combinations to state exactly how large an outcome space is — a 6-from-49 lottery has C(49,6) possible tickets, a number worth knowing before quoting odds, which the Probability Calculator then converts into an actual chance of winning. Security-conscious developers use permutations with repetition to state how large a PIN or code space actually is, the same math behind how the Random Number Generator reports a code space's size. Schedulers and event planners use combinations for committee and team selection, and circular permutations specifically for seating charts around a round table. Students studying genetics reach for multiset permutations directly — counting how many distinct sequences a strand of DNA letters can form is exactly the same problem as counting a word's letter arrangements. Puzzle and game designers lean on the same with-repetition formulas to state how large a combination lock's or a board game's state space actually is, often as a way of proving a puzzle is hard enough to be interesting without being so large it becomes unsolvable in practice.

The Honest Limits of This Calculator

This tool counts arrangements and selections exactly — it does not compute probabilities from those counts, which is a deliberately separate job handled by the Probability Calculator. Extremely large results are truncated for display once they exceed roughly 24 digits, showing the leading and trailing digits alongside the total digit count rather than the full number, though the underlying BigInt calculation itself never rounds or loses precision. And for the plain factorial values underlying every formula here, the Factorial Calculator covers double factorials and derangements in more depth than belongs in a combinations tool.