Combinations and Permutations Calculator
nCr and nPr, with or without repetition — exact answers for huge counts, the formula substituted, and a plain explanation of when order matters.
Combinations and Permutations Calculator: with the default inputs, number of ways is 120.
Blank when the exact value is too large for a number; see the exact text.
- Exact value
- 120
- Formula
- n! ÷ (r! × (n − r)!) = 10! ÷ (3! × 7!)
- Chance of one specific outcome
- 0.833333%
Assumptions
- Counts are computed exactly with arbitrary-precision integers; the numeric output is blank only when the value exceeds about 1.8 × 10³⁰⁸.
- n and r are capped at 1,000.
How this is worked out
The formula
Permutations (order matters): P(n, r) = n! ÷ (n − r)! Combinations (order doesn't matter): C(n, r) = n! ÷ (r! (n − r)!) — also written nCr or "n choose r" With repetition: permutations = nʳ, combinations = C(n + r − 1, r) n! = n × (n − 1) × … × 2 × 1, and 0! = 1
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- What are you counting?
- Choose one of 4 options.Combinations nCr — order doesn't matter, no repeats · Permutations nPr — order matters, no repeats · Combinations with repetition — order doesn't matter, repeats allowed · Permutations with repetition — order matters, repeats allowed (nʳ)
- n — items to choose from
- A number.from 0 to 1000 · whole numbers only · defaults to 10
- r — items chosen
- A number.from 0 to 1000 · whole numbers only · defaults to 3
What you get back
- Number of waysmain answer
- Blank when the exact value is too large for a number; see the exact text.
- Exact value
- Formula
- Chance of one specific outcome
What this assumes
- Counts are computed exactly with arbitrary-precision integers; the numeric output is blank only when the value exceeds about 1.8 × 10³⁰⁸.
- n and r are capped at 1,000.
About this calculator
Counting problems come down to one question: does order matter? Picking a 3-person committee from 10 people is a combination — Ann, Bo, Cy is the same committee as Cy, Ann, Bo. Awarding gold, silver and bronze to 3 of 10 runners is a permutation — the same three people in a different order is a different result. Permutations always outnumber combinations by exactly r!, the number of ways to shuffle the chosen items.
How to use it
Pick the mode, enter n (how many you're choosing from) and r (how many you choose). The answer is exact even when it has hundreds of digits — a 52-card deck has 52! ≈ 8 × 10⁶⁷ orderings and the calculator will tell you every digit — and the steps show the formula with your numbers substituted.
The four cases
- Permutations, no repeats (nPr): n × (n − 1) × … for r factors. Each pick removes an option.
- Combinations, no repeats (nCr): the permutation count divided by r!, because each group of r was counted r! times in different orders. C(10, 3) = 720 ÷ 6 = 120.
- Permutations with repetition (nʳ): each of r slots has all n options again — PIN codes, passwords, dice rolls. Four-digit PINs: 10⁴ = 10,000.
- Combinations with repetition: choosing r scoops from n flavours, repeats allowed. The formula C(n + r − 1, r) comes from the "stars and bars" argument: line up r stars and n − 1 dividers and count the arrangements.
Reading the results
"Chance of one specific outcome" is simply 1 ÷ (number of ways), useful for lottery-style questions: 6 numbers from 49 gives C(49, 6) = 13,983,816 combinations, so one ticket has a 0.0000072% chance.
Common mistakes
- Using nPr for a lottery or a committee. If swapping two chosen items gives the "same" result, use nCr.
- Forgetting that 0! = 1, so C(n, 0) = C(n, n) = 1: there is exactly one way to choose nothing and one way to choose everything.
- Choosing r > n without repetition — impossible, and the calculator says so rather than returning 0 silently.
Frequently asked questions
▸What's the difference between a combination and a permutation?
Order. A permutation counts arrangements (ABC and CBA are different); a combination counts selections (ABC and CBA are the same group). nPr = nCr × r!.
▸How do I calculate 10 choose 3 by hand?
Multiply 3 decreasing numbers from 10 and divide by 3!: (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120.
▸What does 'with repetition' mean?
The same item can be picked more than once. A 4-digit PIN allows repeated digits, so there are 10⁴ = 10,000 of them; without repetition there would be only 10 × 9 × 8 × 7 = 5,040.
▸Why is 0! equal to 1?
So the formulas stay consistent: there is exactly one way to arrange nothing, and C(n, n) = n! ÷ (n! × 0!) must equal 1.
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