Permutation & Combination Calculator
Calculate nPr (permutations) and nCr (combinations) with step-by-step factorial expansion. Supports n and r up to 20.
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How to use this calculator
Permutations count ordered arrangements; combinations count unordered selections. nCr = nPr / r! because each combination corresponds to r! permutations.
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Enter n — the total number of items available.
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Enter r — the number of items to choose.
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Both nPr (ordered arrangements) and nCr (unordered selections) are calculated with factorial steps shown.
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n must be at most 20 to avoid integer overflow.
Frequently asked questions
What is the difference between permutations and combinations?
Permutations care about order — choosing president, vice-president, treasurer from 10 people is a permutation (ABC ≠ BAC). Combinations do not care about order — choosing 3 members for a committee is a combination (ABC = BAC = CAB). nCr = nPr / r! because each group of r people can be ordered in r! ways.
When do I use nPr vs nCr?
Use nPr when order matters: PIN codes, race finishing positions, seating arrangements, passwords. Use nCr when order does not matter: lottery numbers, team selection, poker hand types, committee choices, or any selection problem.
How do factorials grow?
10! = 3,628,800. 15! ≈ 1.3 trillion. 20! ≈ 2.4 × 10^18. Factorials grow faster than exponentials, which is why large combinatorics problems have astronomically many possibilities. The number of ways to shuffle a standard deck of 52 cards is 52! ≈ 8 × 10^67 — more than the number of atoms in the observable universe.
What does C(n, 0) or C(n, n) equal?
Both equal 1. C(n,0) = n!/(0!×n!) = 1 because there is exactly one way to choose nothing. C(n,n) = n!/(n!×0!) = 1 because there is exactly one way to choose everything. This is why every row of Pascal's Triangle starts and ends with 1.
Permutation and combination calculator with factorial steps
Permutations: order matters
nPr = n!/(n−r)! counts the number of ways to arrange r items chosen from n, where the order of selection matters. The formula "uses up" n items one at a time: n choices for the first position, n−1 for the second, …, n−r+1 for the rth. Multiplying these gives n × (n−1) × … × (n−r+1) = n!/(n−r)!. Memorize the mnemonic: Permutation = Position matters.
Combinations: order does not matter
nCr = n!/(r!(n−r)!) counts the number of ways to choose r items from n, ignoring order. Since each set of r items can be arranged in r! orders (all of which are counted separately in nPr), divide nPr by r! to get nCr. nCr appears as row n, position r of Pascal's Triangle (0-indexed) and is read "n choose r." The identity C(n,r) = C(n, n−r) reflects that choosing r items to include is the same as choosing n−r items to exclude.
Applications in probability and statistics
In probability, the number of equally likely outcomes uses permutations and combinations: lottery odds (6 numbers from 49 with no order: C(49,6) ≈ 14 million), poker hands (5-card hands from 52: C(52,5) = 2,598,960), and birthday problem calculations all rely on these formulas. In statistics, the binomial distribution uses C(n,k) as its coefficient, and hypothesis testing for contingency tables uses combinations in Fisher's exact test.
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