How to use the Permutation and Combination Calculator
- 1. Enter n and rTotal items and how many you choose.
- 2. Pick what to countAll four basic rules, circular arrangements, or a word or multiset.
- 3. CalculateSee exact results with the formula used, and copy every digit.
Which rule to use
Ask two questions. Does order matter, and can items repeat? Order matters with no repeats: permutations, nPr = n! ÷ (n − r)!. Order doesn't matter with no repeats: combinations, nCr = n! ÷ (r!(n − r)!). With repeats allowed, ordered selections number nʳ, and unordered ones (C(n + r − 1, r)) use the stars-and-bars formula. Seating n people round a table has (n − 1)! arrangements, because rotations look the same. Arranging letters with repeats divides n! by the factorial of each repeat count.
Worked examples
Choosing 3 of 10 students: a captain, vice-captain and secretary (order matters) can be picked in P(10, 3) = 720 ways, while a plain committee of 3 can be picked in C(10, 3) = 120 ways. A 3-digit code from 10 digits with repeats allowed has 10³ = 1,000 options, and choosing 3 scoops from 10 flavours with repeats allowed gives 220. The letters of MISSISSIPPI (1 M, 4 I, 4 S, 2 P) can be arranged in 11! ÷ (1! × 4! × 4! × 2!) = 34,650 distinct ways.
Exactness
- All arithmetic uses whole-number big integers, so even C(5000, 2500) is exact; copy the full value with Copy.
- Results were checked against Python's math.comb and math.perm on 100 random cases.
- n and r can go up to 5,000.
Frequently asked questions
What's 0!?
1, by definition, which makes C(n, 0) = C(n, n) = 1 work out.
Why are necklaces half of circular arrangements?
A necklace can be flipped over, so clockwise and anticlockwise orders count as the same, halving the count for 3 or more items.
Is the permutation and combination calculator free?
Yes. You can open the permutation and combination calculator and enter everything without signing up. When you calculate for the first time, we ask once for your name and WhatsApp number; after that it, and every other Uncodemy tool, opens straight away.