Permutation & Combination Calculator

About the Permutation & Combination Calculator

Count the ways to arrange or choose things, with the formula, the numbers put into it and the cancellation that leads to the answer: permutations nPr and combinations nCr, both also with repetition, circular arrangements (round tables, or necklaces where turning over gives the same one), arrangements of a word with repeated letters such as MISSISSIPPI — with the vowels together, apart or never side by side, and the word’s place in dictionary order — derangements (no item in its own place) and stars and bars (identical items into boxes, with a minimum or maximum per box).

Every answer is an exact whole number, however large: 52C26 has 15 digits and 1000! has 2,568, all of them shown and downloadable. For small cases the calculator also lists the actual arrangements, and Pascal’s triangle highlights nCr and the two numbers above it that add up to it.

Common uses

• Homework and competitive-exam practice on permutations and combinations, with the working to compare.
• Counting passwords, PIN codes, number plates or lottery tickets.
• Planning seating, teams, tournaments and schedules.
• Probability: the number of equally likely outcomes behind a chance, such as the 2,598,960 five-card poker hands.

Permutation or combination?

Ask whether swapping two of the chosen items gives something different. If it does — first and second prize, a PIN, seats in a row — order matters and you count permutations. If it does not — a team, a hand of cards, toppings on a pizza — you count combinations, which is the number of permutations divided by the r! orders in which the same r items can be listed: nCr = nPr ÷ r!.

Then ask whether an item can be used more than once. Digits in a PIN can repeat (10⁴ = 10,000 PINs of four digits); people in a team cannot.

The formulas

• Permutations: nPr = n! ÷ (n − r)! — with repetition: nʳ.
• Combinations: nCr = n! ÷ (r! (n − r)!) — with repetition: C(n + r − 1, r).
• Circular: (n − 1)!, or (n − 1)! ÷ 2 when turning the circle over gives the same arrangement; r of n around a circle: nPr ÷ r.
• A word of n letters with repeats n₁, n₂, …: n! ÷ (n₁! × n₂! × …).
• Derangements: !n = n! × (1 − 1/1! + 1/2! − … + (−1)ⁿ/n!), the nearest whole number to n!/e.
• Stars and bars: n identical items into k boxes in C(n + k − 1, k − 1) ways, or C(n − 1, k − 1) with none empty.

nPr, nʳ, nCr and n! ÷ (n₁! × n₂! × …) for repeated letters are the formulas of the NCERT Mathematics textbook for Class 11 (chapter “Permutations and Combinations”); circular arrangements, combinations with repetition, derangements and stars and bars are standard results of combinatorics beyond that chapter. 0! = 1, so nC0 = nCn = 1.

Words, vowels and dictionary order

For “all the vowels together”, glue the vowels into one block, arrange the block with the consonants, and multiply by the arrangements of the vowels inside the block — dividing out repeated letters at both stages. “Not all together” is all the arrangements minus those. For “no two vowels side by side”, arrange the consonants first and choose a gap between them for each vowel. The calculator lists the arrangements of short words, so you can check every one of these counts by hand.

The place of a word in dictionary order counts the arrangements that come before it: MOTHER comes 309th of the 720 arrangements of its letters.

Limitations

• Exact digits are worked out for answers of up to 50,000 digits; larger ones are given as a power of ten.
• Lists of arrangements are made for up to 100,000 of them, and 1,000 are shown on the page; the downloaded file has them all.
• Vowels are A, E, I, O and U (Y counts as a consonant). Words can have up to 5,000 letters; the vowel conditions work for up to 3,000, the place in dictionary order for up to 400, and the arrangements are listed for words of up to 100 letters.
• Stars and bars puts identical items into distinct (labelled) boxes. Identical items into identical boxes is a different problem (integer partitions) and is not covered.

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