Permutations with Repetition Calculator

What the permutations with repetition calculator does

A permutation is an arrangement of items where the order counts. This calculator counts how many arrangements are possible when you draw a number of items from a set, in both cases: when each item can be used only once, and when the same item can be used again.

Arrangements multiply fast, and the counts get large in a hurry. Below is how it works and how it differs from a combination.

How to use it

1. Enter the number of objects to choose from, and how many you are arranging.
2. Choose "with repetition" yes or no, depending on whether items can repeat.
3. Press Calculate for the number of permutations, or Reset to clear it.

Permutations mean order matters

In a permutation, rearranging the items makes a genuinely different result. The arrangement A, B, C is not the same as C, B, A. This is what sets a permutation apart from a combination, where order is ignored, and it is why there are always at least as many permutations as combinations for the same choice.

Order matters whenever position carries meaning. A PIN, the finishing order of a race, the seating around a table, the letters in a word. In each, shuffling the items produces something new, which is exactly what a permutation counts.

How the two cases are worked out

With repetition, where each position can be filled by any of the items, the count is simply the number of objects raised to the power of how many you are arranging:

Permutations with repetition = n raised to the power r

Without repetition, each item used removes one from the pool, so the count is a shrinking product worked out from factorials, n factorial divided by (n minus r) factorial. The with-repetition case grows fastest, because nothing is ever used up.

With repetition and without

With repetition fits when the same item can appear again in the arrangement. Each digit of a PIN can be any of the ten digits, including ones already used, so every position has the full set to draw from. This is the case for codes, passwords, and anything where slots are filled independently.

Without repetition fits when each item can be used only once. Arranging people in a line, or awarding distinct medals, uses up each item as it is placed, so the pool shrinks with every choice. Fewer arrangements result than with repetition, because the later positions have less to choose from.

A worked example

Suppose you are forming a code of 4 characters from the 10 digits 0 to 9. If digits can repeat, that is permutations with repetition: 10 raised to the power 4 = 10,000 possible codes, which is why a four-digit PIN has ten thousand possibilities.

If instead every digit must be different, that is without repetition: 10 × 9 × 8 × 7 = 5,040 arrangements. Forbidding repeats roughly halves the count here, because each position after the first has one fewer digit to draw from.

Entering your values

Enter whole numbers for the objects and how many you are arranging. With repetition, the count is the objects raised to the power of the number arranged, which climbs steeply. Without repetition, the count is built from factorials, which grow even faster, so very large inputs go beyond what can be shown exactly.

Questions people ask

What is a permutation?

An arrangement of items where order matters. Changing the order produces a different permutation.

How is it different from a combination?

A permutation counts order, a combination does not. There are always at least as many permutations as combinations for the same selection.

When do I use with repetition?

When the same item can appear more than once, like the digits of a PIN. Then the count is the number of objects raised to the power of how many you arrange.

Why does a four-digit PIN have 10,000 options?

Because each of the four positions can be any of ten digits, and digits can repeat. Ten raised to the power four is 10,000.

References

A quick note on where the methods here come from. Permutations and the counting rules of probability are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering permutations and counting.

1. NIST/SEMATECH e-Handbook of Statistical Methods (counting and permutations). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (counting principles and permutations). https://openstax.org/details/books/introductory-statistics-2e