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Free Online Permutation and Combination Calculator, nPr and nCr

Work out permutations and combinations exactly, with or without repetition. All four counts are shown side by side, because the hard part is not the arithmetic but knowing whether order matters and whether repeats are allowed. Answers stay exact at sizes where an ordinary calculator silently rounds.

Free Forever Nothing Uploaded All Four Counts Exact at Any Size

Free Online Permutation and Combination Calculator, nPr and nCr

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🎲 Permutation and Combination Calculator All four counts at once · exact, however large

The sum runs in your browser. Nothing you type is sent anywhere.

How many to choose from, n

How many you take, r

🎲 Which one do you need?

Two questions decide it. Answer them and the matching count is highlighted on the right.

Does the order matter?

Can the same one be picked twice?

Try one of these

The count

—

The formula—

Digits in the answer—

Chance of one particular result—

All four counts for your n and r. The one matching your two answers is marked.

Order matters?	Repeats?	Formula	Count	The sort of question

Recent counts, kept in this browser only.

Nothing here yet Press Log to history on the first tab to keep a calculation here.

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How to Work Out a Permutation or a Combination

Three steps, and the first one is the whole problem.

Answer the two questions Does the order matter, and can the same thing be picked twice? Those two answers pick the formula, and getting them right is the entire difficulty of this kind of problem.

Type the two numbers How many you are choosing from, and how many you take. For a lottery with six balls from forty nine, n is 49 and r is 6.

Read the count, and glance at the other three The table shows all four so you can see how much difference each answer makes. On 49 and 6 the four answers range from fourteen million to thirteen billion, which is why the choice matters more than the arithmetic.

What to Know Before You Pick a Formula

The two questions that decide which of the four you need.

The two questions are the whole problem. Does the order matter, and can something be repeated. With n of 49 and r of 6 the four answers are 13,983,816, about 1.4 times ten to the ten, 10,068,347,520 and 25,827,165. Same two numbers, answers differing by a factor of a thousand. A calculator that just offers an nPr button and an nCr button has handed you the easy half.

Permutation means order matters, combination means it does not. The three medal places in a race are a permutation: gold, silver and bronze to Ann, Raj and Sam is a different outcome from the reverse. Six lottery balls are a combination: the order they come out does not change your ticket. The words are easy to mix up, so each row of the table says which kind of question it answers.

The answer stays exact at sizes where a calculator rounds. A thousand choose five hundred is a 300 digit number, and this page gives all 300 digits. It does it by multiplying and dividing alternately rather than working out three factorials and dividing, which would build a two and a half thousand digit number on the way to a 300 digit answer and would overflow long before it finished.

Choosing r is the same as leaving out n minus r. Forty nine choose six is the same as forty nine choose forty three, because picking six to take and picking forty three to leave behind are the same decision. That symmetry is why the calculation here always works with the smaller of the two, which keeps it fast and exact even when r is large.

Repetition with order is just n to the power of r. A four digit PIN has ten choices for each of four positions, so ten to the fourth, ten thousand. That is the simplest of the four and the one people get right without help. The awkward one is repetition without order, where the formula is a combination in disguise: n plus r minus one, choose r.

Choosing none has exactly one answer, not zero. There is one way to choose nothing from six things, which is to choose nothing, so six choose zero is one. The same reasoning makes six choose six equal one: there is only one way to take all of them. Both are correct and both look wrong at first glance.

Key Features & Capabilities

What it does, and what it deliberately does not.

All four counts at once Order and repetition as two plain questions, with every combination of answers shown and the matching one marked.

Exact at any size Multiplying and dividing alternately rather than through factorials, so 1000 choose 500 gives all 300 digits.

The kind of question each answers A race, a PIN, a lottery ticket, a scoop of ice cream. The example is usually faster than the definition.

The odds, not just the count One over the count, written as a chance, which is what you actually wanted if the question was about a lottery.

The formula written out With your own numbers in it, so the answer can be checked rather than merely accepted.

Nothing is uploaded All the big number arithmetic happens in this tab and it works offline.

About Permutations and Combinations

Counting problems have a reputation for being hard, and the reputation is earned in a specific place. The arithmetic is not hard. What is hard is deciding which of four formulas applies, and every calculator that offers an nPr button next to an nCr button has quietly left you to do that part alone.

There are only two questions, and both are yes or no. Does the order matter? Can the same thing be chosen more than once? Those two answers pick one of four counts, and the four are not close to each other. With forty nine things and six picks they range from about fourteen million to about fourteen billion. So this page asks the two questions in plain words, shows all four answers side by side, and marks the one your answers point to, with an example of the sort of problem each is for.

The three that catch people out are worth naming. Order with repetition is the easy one, n to the power of r, which is a PIN code. Order without repetition is nPr, which is the first three finishers in a race. No order and no repetition is nCr, the lottery ticket. The fourth, no order but repetition allowed, is the one almost nobody remembers: three scoops of ice cream from ten flavours where you may repeat a flavour and the order in the bowl does not count. Its formula is a combination in disguise, n plus r minus one choose r, and it is 220 rather than the 120 or 1000 you might guess.

On the arithmetic there is one decision worth explaining. The textbook formula for a combination is a factorial divided by two factorials, and following it literally is a bad idea: a thousand choose five hundred would build a two and a half thousand digit number, then another, then divide, and an ordinary calculator would have overflowed several steps earlier. Multiplying and dividing in alternation keeps every intermediate value no larger than the answer itself, which is 300 digits, so the result comes out exactly and quickly.

Frequently Asked Questions

Order, repetition, lottery odds and the awkward cases.

Whether the order counts. The three medal positions in a race are a permutation, because first and second are different outcomes. Six lottery numbers are a combination, because the order they are drawn in does not change your ticket. Permutations are always the larger number.

Ask whether the thing you picked goes back. Lottery balls do not come back, so no repetition. Digits in a PIN can repeat, because using a 7 does not use up all the sevens. If you are unsure, the table shows both and the difference is usually obvious once you see the two numbers.

There are 13,983,816 possible tickets, so one in about fourteen million for a single ticket. The odds row shows that for whatever numbers you type. It assumes every ticket is equally likely, which is true of the draw itself.

Because there is exactly one way to choose nothing: choose nothing. The same logic makes choosing all of them equal one, since there is only one way to take everything. Both results are correct and both surprise people.

n plus r minus one, choose r. It is the least memorable of the four and it comes up more often than you would think, in anything where you are picking quantities of types rather than individual items. Three scoops from ten flavours with repeats allowed is 220.

Yes. A thousand choose five hundred comes out in full, all 300 digits, and it stays exact. The calculation multiplies and divides alternately rather than going through factorials, which is both faster and the reason it does not overflow.

No. The big number arithmetic uses your own browser, so nothing you type leaves the page and it keeps working offline.