Permutation Without Replacement Calculator

This calculator works out the number of ways to choose and arrange a set number of items from a larger collection, where each item can be used only once and the order of selection matters. You enter the total number of items (n) and the number you are selecting (r), and you get the exact count of possible ordered arrangements. The result is the permutation value, often written as nPr or P(n,r).

Formula

The permutation without replacement formula is:

P(n, r) = n! / (n − r)!

Where:

• n = total number of distinct items available (unitless count)
• r = number of items you select and arrange (unitless count, r ≤ n)
• ! = factorial, meaning multiply all positive integers from 1 up to that number (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)

The result is a whole number representing the count of distinct ordered sequences. For example, P(5, 2) = 5! / 3! = 120 / 6 = 20, meaning there are 20 ways to pick and order 2 items from 5.

Worked Example

Using the default values from the calculator above (n = 10, r = 3):

1. Compute 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
2. Compute (10 − 3)! = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
3. Divide: 3,628,800 / 5,040 = 720

So P(10, 3) = 720. The calculator gives the same result: 720. This means if you have 10 distinct books and want to arrange 3 of them on a shelf, there are 720 different possible orders.

When to Use Permutations Without Replacement

Use this calculation when you are selecting items from a set and the following two conditions hold:

• No repetition — once an item is chosen, it cannot be chosen again (without replacement).
• Order matters — swapping two chosen items gives a different outcome (e.g., first place vs. second place).

Common real-world examples include:

• Ranking a top 3 from a group of contestants
• Creating a password using distinct characters where sequence matters
• Assigning different roles (president, treasurer, secretary) to people
• Arranging a subset of books on a shelf

If order does not matter, you need combinations (nCr), not permutations. If items can be reused, you need permutations with replacement, which is simply n^r.

How to Read the Result

The output is a count of possible arrangements. It is always an integer. For example, P(10, 3) = 720 means there are 720 distinct sequences. The number grows very quickly as n or r increases. For instance, P(20, 5) = 1,860,480, and P(50, 10) is over 3.7 × 10^16.

Be careful with the factorial notation: 0! is defined as 1, so P(n, 0) = 1 (there is exactly one way to select nothing). P(n, n) = n! because you are arranging all items.

Common Mistakes and Misconceptions

• Confusing permutations with combinations — permutations count ordered arrangements; combinations count unordered groups. For the same n and r, permutations are always larger (unless r = 0 or 1).
• Using n^r instead of nPr — n^r allows repetition, so it is only correct when items can be reused.
• Entering r > n — mathematically, P(n, r) is undefined (or zero) when r exceeds n because you cannot select more distinct items than are available. The calculator will show an error or a nonsensical result; check your inputs.
• Forgetting that order matters — if you are choosing a committee where all members have equal rank, that is a combination problem, not a permutation.
• Rounding errors — the result is always a whole number; if you see a decimal, you likely used the wrong formula or inputs.

Permutations vs. Combinations vs. With Replacement

To choose the right formula, ask two questions:

1. Does order matter? If yes → permutation; if no → combination.
2. Can an item be used more than once? If no → without replacement; if yes → with replacement.

This gives four possibilities:

• Permutation without replacement (this calculator): n! / (n−r)!
• Permutation with replacement: n^r
• Combination without replacement: n! / (r! (n−r)!)
• Combination with replacement: (n+r−1)! / (r! (n−1)!)

For example, choosing 2 letters from {A, B, C}:

• Permutations without replacement: AB, BA, AC, CA, BC, CB → 6
• Permutations with replacement: AA, AB, BA, BB, AC, CA, CC, BC, CB → 9 (3^2)
• Combinations without replacement: AB, AC, BC → 3
• Combinations with replacement: AA, AB, AC, BB, BC, CC → 6

Limitations and Practical Notes

The formula assumes all n items are distinct. If some items are identical (e.g., repeated letters in a word), the count is smaller and requires dividing by factorials of repeated counts. The calculator does not handle that case.

Also, the result can become astronomically large for moderate inputs. For n = 100 and r = 50, the number has about 93 digits. The calculator uses standard integer arithmetic, so it will display the exact value as long as it fits within the browser’s numeric precision (up to about 1.8 × 10^308). Beyond that, it may show scientific notation or an error.

For very large n and r, you may prefer to work with logarithms or approximations, but for most practical problems (like arranging a dozen items or ranking a few candidates), the exact integer is perfectly manageable.

Frequently Asked Questions

What is the difference between permutation and combination?
Permutation counts ordered selections, so AB and BA are different. Combination counts unordered groups, so AB and BA are the same. For the same n and r, permutations are larger by a factor of r!.

Can r be greater than n?
No, in permutations without replacement you cannot select more items than are available. If r > n, the result is undefined (or zero in some conventions). Always check that r ≤ n.

How do I calculate permutations with replacement?
Simply use n^r. For example, with 10 digits and a 4-digit PIN (digits can repeat), there are 10^4 = 10,000 possible PINs. This calculator does not handle that case; use a regular power calculator.

What does 0! mean?
By definition, 0! = 1. This makes the formula work for edge cases like P(n, 0) = 1 (one way to select nothing) and P(n, n) = n! (arrange all items).

Why is my result a decimal?
The permutation count is always a whole number. If you get a decimal, you likely entered a non-integer for n or r, or you used a different formula. Check your inputs.

Does this calculator work for identical items?
No, it assumes all n items are distinct. If you have repeated items (e.g., arranging letters in the word “MISSISSIPPI”), you need to divide by the factorials of the repeated counts.

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in n | Total items (n) | - | 10 | 0
in r | Items selected (r) | - | 3 | 0
out perm | Permutations (nPr) | - | npr(n,r) | 0