Permutation formula (nPr)

The standard permutation formula (without repetition) is:

• nPr = n! / (n − r)!

This works when you’re selecting r items from n items and you cannot reuse an item once chosen. In other words, you’re arranging distinct items.

If repetition is allowed (you can reuse items), the counting is often:

• n^r (n choices for each of r positions)

Example: a 4-digit PIN with digits 0–9 has 10^4 = 10,000 possibilities because each position can repeat digits.

Quick permutation examples

• Example 1: Arranging 3 out of 10 items (no repetition)

  n = 10, r = 3 → nPr = 10! / 7! = 10×9×8 = 720.

• Example 2: Podium winners

  12 runners, 3 podium spots (1st/2nd/3rd). Order matters, no repetition: 12P3 = 12×11×10 = 1320.

• Example 3: Codes with repetition

  6-character code using 26 letters (A–Z), repetition allowed: 26^6 = 308,915,776.

How permutations really work (and why the formula is so fast)

A permutation is one of those ideas that feels “obvious” once you see it, yet it’s also one of the most common places where counting problems go off the rails. The whole concept boils down to one simple question: when I choose items, does the order of my choices create a different outcome?

If the answer is yes, you’re counting arrangements. Arrangements are permutations. If the answer is no, you’re counting groups. Groups are combinations. The distinction sounds small, but it can change the answer by a factor of r! (which gets huge fast).

Here’s a friendly way to visualize it. Suppose you have 10 unique items (maybe 10 different books), and you want to pick 3 books to place on a shelf in a specific order. The first slot on the shelf can be filled by any of the 10 books. Once you place one book, you usually can’t place the same physical book again (that would be repetition), so the second slot now has 9 choices. The third slot has 8 choices. Multiply the choices because they happen in sequence:

• Choices for slot 1: 10
• Choices for slot 2: 9
• Choices for slot 3: 8
• Total: 10 × 9 × 8 = 720

That multiplication pattern is the heart of permutations. Notice something cool: we never had to compute “10!” as a giant number. We only needed the first r factors going downward from n. That’s why many textbooks rewrite the permutation formula like this:

• nPr = n × (n−1) × (n−2) × … × (n−r+1)

That product form is exactly what this calculator uses for the exact mode, because it’s fast and avoids unnecessary huge factorials. But you might wonder: how does this connect to the classic n! / (n−r)! expression?

Factorials are just shorthand for long products. For example: 10! = 10×9×8×7×6×5×4×3×2×1. And 7! = 7×6×5×4×3×2×1. If you divide 10! by 7!, everything from 7 down to 1 cancels out, leaving: 10×9×8. Exactly the permutation product above. That cancellation is the whole trick.

Now let’s talk about the “with repetition” option, because it shows up in real life all the time. Repetition means that after you choose something for one position, you’re still allowed to choose it again later. Imagine a 4-digit PIN. Even if you already used the digit 7 in the first position, you can still use 7 again in the second or third position. Each of the 4 positions has 10 independent choices (0–9), so the count is 10×10×10×10, which is 10^4.

The difference between repetition and no repetition is a difference in whether the number of choices decreases as you fill positions. Without repetition: choices shrink (n, n−1, n−2, …). With repetition: choices stay constant (n, n, n, …).

Because permutations grow extremely quickly, you’ll often see truly massive results even with medium inputs. That’s normal. In fact, “fast growth” is one of the reasons permutations are so useful in probability and statistics: once you know how many outcomes are possible, you can compute likelihoods by taking favorable outcomes over total outcomes.

A practical example: suppose a teacher randomly assigns 3 distinct prizes to 3 different students out of a class of 25. The order matters because prize #1 and prize #2 aren’t the same. So the count is 25P3: 25×24×23 = 13,800 possible award outcomes. If instead the teacher is choosing 3 students who will each receive the same identical certificate, order does not matter and the count becomes a combination, 25C3.

Another classic use case is ranking. If you are picking a 1st-place winner, 2nd-place winner, and 3rd-place winner from n competitors, you are always doing permutations because each position is a different outcome. That is literally what permutations were born to model: “arrangements in order.”

Finally, if you’re using this tool to double-check homework, here’s a small checklist:

• Step 1: Identify n (total options) and r (positions/choices you are filling).
• Step 2: Ask “does order matter?” If yes → permutation.
• Step 3: Ask “can I repeat items?” If yes → with repetition (often n^r).
• Step 4: Compute using product form or factorial form, then simplify.

If you want a quick memory trick: permutations are about positions (1st, 2nd, 3rd…), combinations are about piles (just a set of items). Positions care about order; piles don’t. Once you internalize that, you’ll spot the right tool instantly.

Permutation Calculator FAQs

• What does nPr mean?

  nPr means “the number of permutations of r items chosen from n items.” It counts how many ordered arrangements are possible when you choose r items out of n and the order matters.

• When should I use permutations instead of combinations?

  Use permutations when different orders count as different outcomes. For example, choosing gold/silver/bronze winners is a permutation problem because (A,B,C) is not the same as (B,A,C).

• What if r is larger than n?

  In the “without repetition” case, r cannot exceed n because you can’t pick more distinct items than exist. This calculator will show an error if r > n in that mode. In “with repetition” mode, r can be larger than n.

• Why are the numbers so huge?

  Permutations grow fast because you multiply several large values together. Even 50P6 is massive. That’s normal in counting theory — and it’s exactly why scientific notation can be helpful.

• Does the calculator use factorials?

  Internally, it uses the product form (n × (n−1) × … × (n−r+1)) for exact results. That is equivalent to n!/(n−r)!, but more efficient and avoids unnecessary huge computations.

• How accurate is the scientific notation mode?

  Scientific notation mode uses logarithms to estimate extremely large results when exact BigInt output would be too large or slow. For normal classroom sizes, exact mode is typically fine.