Permutation Calculator (nPr)
This permutation calculator works out nPr — the number of ways to arrange r items chosen from n, when the order they end up in matters.
Type 10 and 3 and the answer is 720. Not because a formula was applied to it, but because the first place has ten candidates, the second has the nine still standing, and the third has eight: 10 × 9 × 8.
What nPr Is Actually Counting
Almost every permutation question is really the same story told with different props. There is a pool of things. You take them one at a time and put each one somewhere. Nothing goes back.
Eight runners, three medals. The gold can go to any of the eight. The silver can go to any of the seven still without a medal. The bronze, to any of the six left. Multiply and you have 8P3 = 336 possible podiums.
The word that decides everything is order. If moving two of your picks around gives a different answer to the question, you are counting arrangements, and this is the tool. If it gives the same answer, you want a selection instead, and the number for that is on screen too — more on it below.
The Permutation Formula, and Why Most of It Cancels
The textbook writes it as a ratio of two factorials:
nPr = n! ÷ (n − r)!
That is honest but wasteful. Counting the arrangements of all ten items gives 3,628,800, far more than we want, so the arrangements of the seven we never placed are divided back out. Every factor from seven downwards appears on the top and on the bottom, and cancels. What survives is the three factors we started with: 10 × 9 × 8.
This matters for more than tidiness. Building 1000! in order to divide by 500! means handling a number with 2,568 digits to reach one with 1,434. Multiplying the surviving factors directly is the same answer with a fraction of the work, and it is what happens under this page.
Telling a Permutation Question From the Other Kind
Students rarely lose marks on the arithmetic here. They lose them by counting the wrong thing, and the giveaway is usually a single verb in the question.
| If the question says | Then | Use |
|---|---|---|
| arranged, seated, lined up, ranked | positions are distinct | nPr |
| first, second, third; president, treasurer | the roles differ | nPr |
| chosen, selected, a team, a committee | the group is all that matters | nCr |
| a PIN, a code, dialled, rolled | repeats are allowed | nr |
A useful test when the wording is slippery: write out one answer, swap two of its entries, and ask whether the question would now have a different answer. Gold-silver-bronze changes. A three-person committee does not.
Where an Ordinary Calculator Gives Up
Ask a phone or a spreadsheet for the permutation 100P50 and you will most likely get something like 3.0685e+91. That is not a different answer, it is the same answer with 79 digits thrown away, because these tools store numbers as decimals carrying roughly fifteen significant figures.
For a homework answer, fifteen figures are plenty. For a proof, a programming problem or a counting argument you intend to check, they are not. This page never converts to a decimal at any stage — every count is built and kept as a whole number — so 100P50 comes back with all 94 digits, beginning 306851875625 and ending in eight zeros.
Those trailing zeros are not an artefact. They are the factors of 10 collected along the way, and they are a quick sanity check that an answer has not been rounded behind your back. Monkza's scientific calculator carries an nPr key for when you want the number in the middle of a longer sum; this page is for when you want the working, and for when the answer is longer than a display.
The Two Extra Numbers Under the Answer
Below the main result sit two more figures, and they are there because they are the two counts most often mistaken for a permutation.
If the order did not matter. This is nCr, the number of selections rather than arrangements. For 10 and 3 it is 120, exactly a sixth of 720, because each of those 120 groups can itself be ordered in 3! = 6 ways. That relationship, nPr = nCr × r!, is the whole difference between the two ideas written as one line.
If items could be reused. A four-digit PIN may read 7-7-7-7, so nothing ever leaves the pool and every place keeps all ten options: 104 = 10,000. The permutation count for the same two numbers is 10P4 = 5,040. The gap is not small, and choosing the wrong one of the two is a more expensive mistake than any slip in multiplication.
Three Permutation Results That Look Like Errors
Three permutation answers reliably make people think the tool has broken. None of them is wrong.
nP0 = 1. Filling no places sounds like zero ways, but doing nothing is a way of doing it, and it is the only one. Counting it as one is what makes 0! = 1 necessary rather than strange.
nPn = n! Use every item and there is nothing left over to divide away, so 6P6 is 720, which is 6!. The page says so on screen when it happens.
r larger than n. Asking to arrange five items out of three has no answer, because the fourth pick has an empty pool. Rather than returning 0 and leaving you guessing, this page names the problem and points out that if repeats were allowed the question would have an answer after all.
How to Use This Permutation Calculator
Put the size of the pool in the first box and the number of places in the second, then press =. The keypad is built into the page, so nothing depends on your phone's keyboard, and the ⇆ key swaps the two boxes for the very common case of typing them the wrong way round.
Show the working opens the cancellation, the surviving factors written out, and the same two numbers asked the other two ways. Below that, the step panel explains the reasoning in words with the sums carried through. Six examples run on a tap, and one of them is deliberately impossible so you can see how the page handles a question that has no answer.
Everything is checked before it is shown: each result is put back through nPr = nCr × r!, Pascal's rule and the symmetry nCr = nC(n−r), and the count of those checks is printed with the working.
Permutation Calculator FAQ
What does nPr mean?
What is the formula for a permutation?
What is the difference between a permutation and a combination?
How do you work out 10P3?
Why is nP0 equal to 1 and not 0?
Is nPn the same as n factorial?
Can the same item be used twice in a permutation?
Why does my calculator show 3.06e+91 instead of the answer?
Count the places, not the possibilities — every permutation is just a pool shrinking by one.