Permutations Calculator
Ordered arrangements, counted properly
Counting arrangements matters more than most people realize, from ranking contest entries to planning delivery routes. A permutations calculator counts the ordered ways to arrange k items taken from a pool of n, which is why teachers, students, event planners, and security analysts use it constantly. Order is the defining feature: the sequence blue, red, green differs from green, red, blue, and a permutation tool counts both.
If order does not matter, you need a combinations calculator instead. Swapping those two questions is the most common mistake students make with these formulas.
The formula with real numbers
The formula is P(n,k) = n!/(n-k)!, which multiplies the k largest factors of n factorial. With 8 candidates competing for 3 distinct medals, the count is 8 x 7 x 6 = 336, because the first slot offers 8 choices, the second only 7, and the third just 6. The classic 4 digit PIN works the same way when repetition is allowed: 10 x 10 x 10 x 10 = 10000 possible codes.
Turn on the With repetition option whenever selected items can appear more than once, like PIN digits or jersey numbers. Leave it off for races, seat assignments, and elimination draws where each item can appear only once.
Using the calculator
Using the tool takes seconds. Type your pool size into Choose from, enter the arrangement length in How many?, set the With repetition toggle to match your scenario, and press Calculate.
The result appears instantly, so arranging 50 items taken 3 at a time returns 117600. A quick sanity check helps: in without-repetition problems the factors step down one at a time, while with repetition every factor equals the pool size, and counts climb steeply as the pool grows.
Frequently Asked Questions
How many 4 digit PINs are possible?
With ten digits 0 through 9 and repetition allowed, each position has 10 choices, so 10 x 10 x 10 x 10 equals 10000 PINs. That is why four-digit codes are considered weak against guessing attacks.
What is the difference between permutations and combinations?
Permutations count orders, so ABC and CAB are different, while combinations treat them as identical picks. Choosing 3 of 8 people for ordered roles gives 8 x 7 x 6 = 336 permutations but only 56 combinations.
What does the with repetition option change?
Repetition lets an item appear more than once, multiplying choices every step instead of shrinking them. Drawing 3 of 10 with repetition allows 1000 arrangements, while without repetition it allows only 10 x 9 x 8 = 720.
Why does the count drop so quickly?
Each pick permanently removes one item from the pool in the without-repetition case, so the factors shrink: choosing 5 from 20 gives 20 x 19 x 18 x 17 x 16 = 1860480, which falls fast if you lower either number.
What happens when I ask for too many?
If the number you request exceeds the pool without repetition, no arrangement exists and the answer is 0. For instance you cannot order 6 distinct items from a pool of 4, so the calculator returns zero rather than a count.