Combinations & Permutations Calculator

Order. Combinations count selections where order is irrelevant: picking Alice and Bob is the same team as picking Bob and Alice, so C(5, 2) = 10. Permutations count ordered arrangements: Alice-then-Bob differs from Bob-then-Alice, so P(5, 2) = 20. The permutation count is always the combination count times r!, the number of ways to shuffle each selection.

Use C(n, r) = n! / (r! × (n - r)!). For example, C(5, 2) = 120 / (2 × 6) = 10. In practice you cancel before multiplying: C(5, 2) = (5 × 4) / (2 × 1), which keeps the numbers small. The calculator shows this expansion for small n and switches to grouped or scientific form when the factorials get enormous.

Two different formulas take over. Ordered with repetition is n^r: a 4-digit PIN has 10^4 = 10,000 possibilities. Unordered with repetition uses the stars and bars formula C(n + r - 1, r): choosing 3 scoops from 4 ice-cream flavors gives C(6, 3) = 20 ways. Switch the repetition option and the calculator applies the right formula and names it.

Without repetition, each choice uses an item up, so you cannot pick 6 distinct items from a set of 5. With repetition allowed there is no such limit for ordered picks, since every slot resets to all n choices. The calculator explains this instead of returning a silent zero.

Exact. The arithmetic runs on big integers, not floating point, so C(1000, 500) comes out to every last digit rather than an approximation. For readability the calculator also shows a scientific form such as 2.70 × 10^299 alongside the full number.

nCr reads "n choose r": the number of ways to pick r items from n when order is irrelevant, also written C(n, r) or as the binomial coefficient. Its sibling button nPr counts ordered picks. So 5C2 = 10 teams of two from five people, while 5P2 = 20 ordered pairs. Enter the same n and r here and the tool shows both counts side by side with the formula worked.

If digits may repeat and order matters (a PIN), it is 10^4 = 10,000, which is technically a permutation count even though people say "combinations". If each digit may appear once, it drops to P(10, 4) = 5,040. And if order truly did not matter, C(10, 4) = 210. The everyday phrase hides three different questions, which is exactly what the order and repetition switches untangle.

n! ("n factorial") is the product of every whole number from n down to 1: 5! = 5 × 4 × 3 × 2 × 1 = 120, and by definition 0! = 1. It counts the ways to arrange n distinct items in a row, which is why both counting formulas are built from factorials. They grow explosively: 20! already exceeds 2.4 × 10^18, which is why this tool computes them with exact big integers.

No. The counting runs entirely in your browser. Nothing is sent to a server, there is no account, and the calculator works offline once the page has loaded.