Probability
Permutation Calculator
Compute the exact number of permutations nPr as an integer, and show the falling product inside n factorial over (n minus r) factorial.
Worksheet
Press Calculate to see the result.
Compute the exact number of permutations nPr as an integer, and show the falling product inside n factorial over (n minus r) factorial.
What this calculator answers
A permutation counts ordered arrangements. If 10 runners can fill 3 podium places, the gold, silver, and bronze choices are ordered, so the count is 10 × 9 × 8 rather than a combination. This page computes that falling product as an exact integer. The same integer is n factorial divided by (n − r) factorial, and the steps write that fraction so the cancellation is visible. Order matters here. If the podium places were an unordered shortlist, the combination page would be the right one.
nPr is computed by the exact permutation routine, which multiplies n × (n − 1) × … × (n − r + 1). The steps also write the factorial form n! / (n − r)!. Choosing r = 0 returns 1, the empty arrangement. Choosing r = n returns n factorial. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Permutations of n items taken r at a time uses this relationship.
nPr = n! / (n − r)! = n × (n − 1) × … × (n − r + 1)
- n. How many distinct items are available.
- r. How many of them are arranged, in order.
- n − r. The items left out of the arrangement.
- nPr. The exact number of ordered arrangements.
There are n choices for the first position, n − 1 for the second, and so on for r positions. That product is exactly n factorial with the unused tail (n − r) factorial cancelled. The cancellation is why the page can multiply r factors instead of computing two huge factorials and dividing. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
n is 10 and r is 3. The formula is n! / (n − r)! = 10! / 7!. The falling product is 10 × 9 × 8.
The exact integer from that product is 720. Order matters. An unordered selection of 3 items out of 10 would be the combination, which divides this count by 3!. The denominator uses (n − r)! = 7!, and 0! is 1 when r equals n.
r = 3 does not exceed n = 10. The result is an exact integer count of arrangements, not a probability.
| nPr | 720 |
|---|---|
| n | 10 |
| r | 3 |
A second situation
Arranging all ten items, which is 10 factorial
n is 10 and r is 10. The formula is n! / (n − r)! = 10! / 0!. The falling product is 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
The exact integer from that product is 3,628,800. Order matters. An unordered selection of 10 items out of 10 would be the combination, which divides this count by 10!. The denominator uses (n − r)! = 0!, and 0! is 1 when r equals n.
r = 10 does not exceed n = 10. The result is an exact integer count of arrangements, not a probability.
| nPr | 3,628,800 |
|---|---|
| n | 10 |
| r | 10 |
Arranging all n items sets r equal to n, so the denominator is 0 factorial, which is 1, and nPr collapses to n factorial. Arranging only r of them stops the product early. The formula name does not change. Only the length of the product changes. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
r moved from 3 to 4, so one more factor entered the falling product. The headline result went from 720 to 5,040.
The nudged inputs
n is 10 and r is 4. The formula is n! / (n − r)! = 10! / 6!. The falling product is 10 × 9 × 8 × 7.
The exact integer from that product is 5,040. Order matters. An unordered selection of 4 items out of 10 would be the combination, which divides this count by 4!. The denominator uses (n − r)! = 6!, and 0! is 1 when r equals n.
r = 4 does not exceed n = 10. The result is an exact integer count of arrangements, not a probability.
| nPr | 5,040 |
|---|---|
| n | 10 |
| r | 4 |
How to read the result
Read the result as a count of ordered lists, not as a probability. Divide by a denominator of possible outcomes only if you are building a probability from this count. The integer is exact. It is not rounded. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The n items are distinct.
- Order matters.
- r is a whole number from 0 through n.
- Items are not repeated inside one arrangement.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Using nPr when the shortlist is unordered. Divide by r factorial, which is the combination.
- Allowing r larger than n. You cannot arrange more items than you have.
- Computing n factorial and (n − r) factorial in floating point and dividing. Large values lose digits. This page keeps the integer.
- Treating nP1 as 1. There are n choices for a single ordered slot.
Where a different calculator is the honest choice
Use the combination calculator when order does not matter. Use the factorial calculator when r equals n and you only want n factorial. Use the binomial calculator when the arrangement count is only one piece of a success-and-failure probability. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Combination Calculator, Factorial Calculator, Binomial Distribution Calculator.
Check yourself
How many ordered ways can 10 distinct items fill 3 positions?
10P3 = 10! / 7! = 10 × 9 × 8 = 720.
Using the result in a sentence
Podium places, password positions drawn from distinct characters, and seating charts are ordered-arrangement questions. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Identical items, or arrangements that allow repetition, need a different count. This integer also grows so fast that a probability built from two such counts should usually be computed in log space rather than by dividing the displayed integers in a spreadsheet. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Permutation Calculator commit to?
This page uses Permutations of n items taken r at a time: nPr = n! / (n − r)! = n × (n − 1) × … × (n − r + 1). There are n choices for the first position, n − 1 for the second, and so on for r positions. That product is exactly n factorial with the unused tail (n − r) factorial cancelled. The cancellation is why the page can multiply r factors instead of computing two huge factorials and dividing. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for n items, r arranged, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the npr is 720. Read the result as a count of ordered lists, not as a probability. Divide by a denominator of possible outcomes only if you are building a probability from this count. The integer is exact. It is not rounded. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Using nPr when the shortlist is unordered. Divide by r factorial, which is the combination. Identical items, or arrangements that allow repetition, need a different count. This integer also grows so fast that a probability built from two such counts should usually be computed in log space rather than by dividing the displayed integers in a spreadsheet. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the combination calculator when order does not matter. Use the factorial calculator when r equals n and you only want n factorial. Use the binomial calculator when the arrangement count is only one piece of a success-and-failure probability. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
10P3 = 10! / 7! = 10 × 9 × 8 = 720. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.