The Permutation Calculator finds the number of ordered arrangements you can make when choosing $r$ items from $n$ distinct items. It supports both without repetition (no item reused) and with repetition (items can be reused).
What does the calculator do?
Permutations without repetition (order matters) — $n\mathrm{P}r$:
$$
P(n,r) = \frac{n!}{(n-r)!}
$$
Permutations with repetition (order matters, reuse allowed):
$$
n^r
$$
Validates inputs (non-negative integers; for no-repetition, requires $r \le n$).
Formats large results (scientific notation if needed) and provides a copy button.
Worked examples
Example 1 — Without repetition
How many 3-digit codes using digits $0$–$9$ without reusing a digit? ($n=10$, $r=3$)
$$
P(10,3) = \frac{10!}{(10-3)!} = 10 \times 9 \times 8 = 720
$$
Example 2 — With repetition
How many 3-digit codes if digits can repeat? ($n=10$, $r=3$)
$$
10^3 = 1000
$$
Example 3 — Letters without repetition
How many 4-letter arrangements from the 26 letters, no repeats? ($n=26$, $r=4$)
$$
P(26,4) = 26 \times 25 \times 24 \times 23 = 358{,}800
$$
Example 4 — Edge cases
- If $r=0$, there’s exactly one arrangement (the empty arrangement):
$$
P(n,0) = 1
$$
- If $r>n$ and no repetition allowed, it’s impossible: result is undefined (the calculator will show an error).
Why is it important?
- In school: Builds core counting-principle intuition before combinations and binomial topics.
- In real life: PINs, passwords, seatings, race orders, license plates.
- In STEM: Foundations for probability, cryptography, and algorithm analysis.
Frequently Asked Questions
Q1: What’s the difference between permutations and combinations?
Permutations care about order; combinations don’t. Permutations without repetition are calculated as $P(n,r) = \frac{n!}{(n-r)!}$, while combinations are $C(n,r) = \frac{n!}{r!(n-r)!}$.
Q2: When do I use $n^r$ instead of $P(n,r)$?
Use $n^r$ when repetition is allowed; use $P(n,r)$ when no repeats are allowed.
Q3: Why does the calculator sometimes say the number is “too large”?
JavaScript Number overflows around $170!$. For very large $n$ and $r$, we display scientific notation or indicate it’s too large for exact Number arithmetic.
Q4: Can I use decimals?
No. Permutations are only defined for whole counts. Enter non-negative integers.