math calculator

Permutation Calculator — Calculate nPr & Combinations Instantly

Calculate permutations (nPr) and combinations (nCr) instantly with step-by-step work. Perfect for seating arrangements, passwords, lottery odds, card hands, and more.

Formula Verified by Dr. Elena Rostova, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Select Mode

Inputs

Total Items (n)

Chosen Items (r)

Allow repetition of items

Active Output Display

Results

Permutations Result

504

Permutations (P)504

Combinations (C)84

Permutation Step-by-Step

9P3 = 9! / (9 - 3)!

= 9! / 6!

9 × 8 × 7

= 504

Combination Step-by-Step

9C3 = 9! / (3!(9 - 3)!)

= (9 × 8 × 7) / 3!

= 504 / 6

= 84

Factorials Reference

n! (9!) = 362,880

r! (3!) = 6

(n - r)! (6!) = 720

Bookmark this permutation calculator

Save this page for quick access when working on combinatorics homework, probability problems, exam prep, or real-world planning like seating charts and password policies.

Formula

Permutation & Combination Formulas

Without repetition:
  nPr = n! / (n - r)!
  nCr = n! / (r!(n - r)!)

With repetition:
  nPr (rep) = n^r
  nCr (rep) = C(n + r - 1, r)

Permutations count ordered arrangements where order is critical — swapping two items creates a new result. Combinations count unordered selections where only the group of items matters, not their order.

Step by step

How the calculation works

1Enter the total number of items (n) and how many you're choosing (r).
2Toggle "Allow repetition" on if items can be reused (like PIN digits); leave it off for problems like seating people or dealing cards.
3Pick which output to highlight — Permutations (nPr) or Combinations (nCr) — the calculator computes both either way.
4Review the step-by-step factorial breakdown, the factorials reference card, and (for small sets) the full list of generated subsets.
5Switch to the Real-World Scenarios tab to explore the same math applied to seating charts, passwords, lotteries, card hands, races, committees, and locks.

Example

Combinatorics Example — Choosing and Arranging 3 Candidates from 9

Choosing 3 candidates from 9 for a committee: Combinations = 9! / (3! × 6!) = 84 ways. Arranging those same 3 candidates into President, Vice President, and Secretary roles: Permutations = 9! / 6! = 504 ways. Same items, different question — that's the core difference between nCr and nPr.

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Combination

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Calculate combinations (nCr) for problems where order doesn't matter — lottery, teams, and sampling.

Factorial

Factorial & Gamma Calculator

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Generate random numbers within a custom range, with or without repeats.

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Compute mean, median, mode, variance, and standard deviation for a dataset.

Permutations vs Combinations Growth

Permutations (ordered choices) scale much faster than combinations (unordered choices) as the selection size increases.

How to understand and use this calculator

Quick Answer

What is a permutation? A permutation is the number of ways to arrange r items from a set of n distinct items where order matters.

Formula:

nPr = n! / (n - r)!

Simple example: How many ways to arrange 3 cards from a deck of 9?

9P3 = 9! / (9-3)! = 9! / 6! = 9 × 8 × 7 = 504

Key difference:

Permutations: Order matters (arrangements)
Combinations: Order doesn't matter (selections)

Quick reference:

5P2 = 5 × 4 = 20
5C2 = 5 × 4 / 2 = 10
10P3 = 10 × 9 × 8 = 720
10C3 = 10 × 9 × 8 / 6 = 120

Use the Permutation Calculator

To use the nPr calculator:

1. Enter the total number of items (n).

2. Enter the number of items to choose (r).

3. Toggle repetition on or off, and pick your output mode.

4. Review the number of permutations (nPr) and the step-by-step work shown below.

Example input: Total Items (n) = 9, Items to Choose (r) = 3.

Output:

Permutations (9P3) = 504
Combinations (9C3) = 84
Step-by-step work: 9P3 = 9!/(9-3)! = 9!/6! = 9×8×7 = 504

Another example: Total Items (n) = 5, Items to Choose (r) = 2.

Permutations (5P2) = 20
Combinations (5C2) = 10
Step-by-step work: 5P2 = 5!/(5-2)! = 5!/3! = 5×4 = 20

What is a Permutation?

A permutation is the number of ways to arrange a set of items where the order of the items matters.

Think of it like this: if you're lining people up for a photo, the order in which they stand matters. Alice first and Bob second is different from Bob first and Alice second.

Key characteristics:

Items are distinct (different from each other)
Order matters (arrangements)
No repetition allowed (usually — toggle repetition on for cases where it is)

Simple example: How many ways to arrange 2 letters from {A, B, C}?

Permutations: AB, AC, BA, BC, CA, CB = 6 ways
Notice: AB is different from BA!

The permutation formula:

nPr = n! / (n - r)!
Where n = total items, r = items to choose

Permutation vs Combination — What's the Difference?

Feature comparison:

Order: Permutations — order matters. Combinations — order doesn't matter.
Also called: Permutations — arrangements. Combinations — selections.
Example: Permutations — AB ≠ BA. Combinations — AB = BA.
Formula: Permutations — n! / (n-r)!. Combinations — n! / (r!(n-r)!).
Value: Permutations are always larger. Combinations are always smaller.
Used for: Permutations — passwords, seating, rankings. Combinations — lottery, teams, sampling.

Example: Set {A, B, C}, choose 2.

Permutations (order matters): AB, AC, BA, BC, CA, CB = 6
Combinations (order doesn't matter): AB, AC, BC = 3

When to use which:

Passwords → Permutations. Order of characters matters.
Lottery → Combinations. Order of numbers doesn't matter.
Seating arrangements → Permutations. Where people sit matters.
Choosing a team → Combinations. Who's on the team, not their order.
Race rankings → Permutations. First, second, third matters.
Picking flavors → Combinations. Order of flavors doesn't matter.

The Permutation Formula (nPr)

The permutation formula calculates the number of ways to arrange r items from n distinct items.

Formula:

nPr = n! / (n - r)!

Where:

n = total number of items
r = number of items to arrange
! = factorial (multiply all numbers from 1 to that number)

Expanded form:

nPr = n × (n-1) × (n-2) × ... × (n - r + 1)

Example: 9P3 = 9 × 8 × 7 = 504 (you multiply 3 numbers starting from 9).

Special cases:

nP0 = 1 (choose nothing)
nP1 = n (choose one item)
nPn = n! (arrange all items)
0! = 1 (mathematical convention)

How to Calculate Permutations — Step by Step

Calculating permutations is simple when you follow these steps.

Step 1: Identify n (total items) and r (items to choose). Example: 9 items, choose 3 → n = 9, r = 3.

Step 2: Write the formula: nPr = n! / (n-r)!

Step 3: Plug in the numbers: 9P3 = 9! / (9-3)! = 9! / 6!

Step 4: Expand the factorial (or multiply): 9! / 6! = 9 × 8 × 7 × 6! / 6! — the 6! cancels out — 9 × 8 × 7 = 504.

Answer: 9P3 = 504

More examples:

5P2 = 5!/(5-2)! = 5×4 = 20
6P3 = 6!/(6-3)! = 6×5×4 = 120
10P4 = 10!/(10-4)! = 10×9×8×7 = 5040
7P2 = 7!/(7-2)! = 7×6 = 42
8P4 = 8!/(8-4)! = 8×7×6×5 = 1680

How to Calculate Combinations (nCr)

A combination is the number of ways to choose r items from n items where order does not matter.

Formula:

nCr = n! / (r!(n - r)!)

Example:

9C3 = 9! / (3!(9-3)!) = 9! / (3! × 6!)
9C3 = (9 × 8 × 7) / (3 × 2 × 1)
9C3 = 504 / 6 = 84

Relationship between permutations and combinations: nCr = nPr / r!

Example: 9P3 = 504, 9C3 = 504 / 6 = 84 ✅

More examples:

5C2 = 5!/(2!×3!) = (5×4)/(2×1) = 10
6C3 = 6!/(3!×3!) = (6×5×4)/(3×2×1) = 20
10C4 = 10!/(4!×6!) = (10×9×8×7)/(4×3×2×1) = 210
7C2 = 7!/(2!×5!) = (7×6)/(2×1) = 21

Real-World Applications of Permutations and Combinations

1. Seating Arrangements (Permutations). Problem: 5 people at a round table — how many seating arrangements? Solution: (5-1)! = 24 (circular permutations). Why: order matters, people are distinct. Try the Real-World Scenarios tab above to explore this interactively.

2. Passwords (Permutations). Problem: how many 4-digit passwords using digits 0-9 without repetition? Solution: 10P4 = 10 × 9 × 8 × 7 = 5040. Why: order of digits matters for passwords.

3. Lottery (Combinations). Problem: pick 6 numbers from 49 in a lottery. Solution: 49C6 = 13,983,816. Why: order doesn't matter — {1,2,3,4,5,6} is the same as {6,5,4,3,2,1}.

4. Card Games (Combinations). Problem: how many 5-card poker hands from a 52-card deck? Solution: 52C5 = 2,598,960. Why: order of cards in your hand doesn't matter.

5. Race Rankings (Permutations). Problem: 8 runners — how many ways to award gold, silver, bronze? Solution: 8P3 = 8 × 7 × 6 = 336. Why: order matters — first, second, third are different.

6. Choosing a Team (Combinations). Problem: choose 3 people from 10 for a committee. Solution: 10C3 = 120. Why: the order on the committee doesn't matter.

7. Lock Combinations (Permutations). Problem: a 3-digit lock with digits 0-9, no repetition. Solution: 10P3 = 10 × 9 × 8 = 720. Why: order of digits matters for the lock, even though it's called a "combination" lock.

Practice Problems

Problem 1: How many ways can you arrange 3 letters from the word "MATH"? Solution: n = 4 (M, A, T, H), r = 3 → 4P3 = 4 × 3 × 2 = 24.

Problem 2: How many ways can you choose 2 flavors from 5 ice cream flavors? Solution: n = 5, r = 2 → 5C2 = (5 × 4) / (2 × 1) = 10.

Problem 3: A teacher needs to assign 3 different roles from 8 students. How many ways? Solution: n = 8, r = 3 → 8P3 = 8 × 7 × 6 = 336.

Problem 4: How many different 4-digit PINs can be made from digits 0-9 without repetition? Solution: n = 10, r = 4 → 10P4 = 10 × 9 × 8 × 7 = 5040.

Problem 5: A pizza shop offers 10 toppings. How many ways to choose 3 toppings? Solution: n = 10, r = 3 → 10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120.

Answers: 1) 24 — 2) 10 — 3) 336 — 4) 5040 — 5) 120

Common Permutation Mistakes

Confusing permutations and combinations. Mistake: using permutations when order doesn't matter. Fix: ask "Does order matter?" If no → use combinations.
Forgetting to subtract r from n. Mistake: nPr = n! / r! (wrong). Fix: nPr = n! / (n-r)!
Not simplifying the factorial. Mistake: 5P2 = 5! / 3! = 120 / 6 = 20 (correct but more work). Fix: 5P2 = 5 × 4 = 20 (easier!).
Misreading the problem. Mistake: using nCr when the problem asks for nPr. Fix: look for keywords like "arrange," "order," or "positions" → permutations.
Forgetting that 0! = 1. Mistake: 5P0 = 0 (wrong). Fix: 5P0 = 1 (choose nothing, 1 way).
Using wrong values. Mistake: using the wrong numbers for n or r. Fix: carefully read the problem to identify n and r.
Not considering repetition. Mistake: using the no-repetition formula for problems where repetition is allowed. Fix: toggle repetition on — the formula becomes n^r.
Not checking answer reasonableness. Mistake: getting an answer that seems too large or too small. Fix: does the answer make sense? If not, double-check.

What This Calculator Can and Cannot Do

This calculator CAN help with:

Calculating permutations (nPr) and combinations (nCr), with or without repetition
Circular (round-table) and linear seating permutations
Factorial calculations, including very large factorials shown in scientific notation
Step-by-step work shown for every calculation
Understanding order matters vs order doesn't matter
Real-world applications — seating, passwords, lottery, cards, races, committees, and locks — in the Scenarios tab
Both small and very large numbers, with results that stay accurate even past standard number limits

This calculator CANNOT do:

Handle permutations of sets with repeated, indistinguishable items (e.g. arranging the letters of "MISSISSIPPI") — that requires a multinomial formula
Calculate probabilities directly
Replace understanding of the underlying concepts

Final Summary

A permutation calculator is essential for anyone working with arrangements and selections — whether you're a student learning combinatorics, a teacher preparing lessons, or a professional analyzing probabilities.

Remember the key formulas:

Permutations (nPr): n! / (n-r)! — order matters
Combinations (nCr): n! / (r!(n-r)!) — order doesn't matter
Relationship: nCr = nPr / r!

Quick reference:

5P2 = 20 (arrangements)
5C2 = 10 (selections)
10P3 = 720
10C3 = 120

Start calculating now! Just enter your values above and get instant results — completely free.

Related concepts

Factorial

The product of all positive integers up to n, written n!, and the building block of every permutation and combination formula.

Fundamental Counting Principle

If one event can happen in m ways and a second in n ways, both together can happen in m × n ways — the basis of combinatorics.

Combinatorics

The branch of mathematics concerned with counting, arranging, and combining objects, underlying probability and statistics.

Frequently Asked Questions

What is a permutation calculator?+

A permutation calculator finds the number of ways to arrange r items from n distinct items where order matters.

What is the permutation formula?+

The permutation formula is nPr = n! / (n-r)!.

What is the difference between permutation and combination?+

In permutations, order matters (arrangements). In combinations, order doesn't matter (selections).

When do I use permutations?+

Use permutations when the order of items matters, such as seating arrangements, passwords, or rankings.

When do I use combinations?+

Use combinations when the order of items doesn't matter, such as lottery numbers, choosing a team, or picking flavors.

What is nPr?+

nPr is the notation for permutations. It means the number of ways to arrange r items from n distinct items.

What is the relationship between permutations and combinations?+

nCr = nPr / r!. Combinations are smaller because order doesn't matter.

How do I compute permutations?+

To compute permutations: identify n and r, use the formula nPr = n!/(n-r)!, and calculate.

What is 0!?+

0! = 1 (mathematical convention).

Can I use permutations with repetition?+

The standard permutation formula (nPr) is for arrangements without repetition. This calculator also supports repetition — toggle "Allow repetition" and it switches to the n^r formula automatically.

What is the difference between nPr and nCr?+

nPr (permutations) is for arrangements where order matters. nCr (combinations) is for selections where order doesn't matter.

How many ways can I arrange 5 books on a shelf?+

5P5 = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways.

How many ways to choose 3 toppings from 10?+

10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120 ways.

What is a possible combinations calculator?+

A possible combinations calculator calculates the number of ways to choose r items from n items where order doesn't matter (combinations).

Is this permutation calculator free?+

Yes — completely free with no registration required.

Permutation & Combination Formulas

Without repetition: nPr = n! / (n - r)! nCr = n! / (r!(n - r)!) With repetition: nPr (rep) = n^r nCr (rep) = C(n + r - 1, r)

How the calculation works

1Enter the total number of items (n) and how many you're choosing (r).

2Toggle "Allow repetition" on if items can be reused (like PIN digits); leave it off for problems like seating people or dealing cards.

3Pick which output to highlight — Permutations (nPr) or Combinations (nCr) — the calculator computes both either way.

4Review the step-by-step factorial breakdown, the factorials reference card, and (for small sets) the full list of generated subsets.

5Switch to the Real-World Scenarios tab to explore the same math applied to seating charts, passwords, lotteries, card hands, races, committees, and locks.

Permutations vs Combinations Growth

Permutations (ordered choices) scale much faster than combinations (unordered choices) as the selection size increases.

How to understand and use this calculator

Quick Answer

What is a permutation? A permutation is the number of ways to arrange r items from a set of n distinct items where order matters.

Formula:

nPr = n! / (n - r)!

Simple example: How many ways to arrange 3 cards from a deck of 9?

9P3 = 9! / (9-3)! = 9! / 6! = 9 × 8 × 7 = 504

Key difference:

Permutations: Order matters (arrangements)
Combinations: Order doesn't matter (selections)

Quick reference:

5P2 = 5 × 4 = 20
5C2 = 5 × 4 / 2 = 10
10P3 = 10 × 9 × 8 = 720
10C3 = 10 × 9 × 8 / 6 = 120

Use the Permutation Calculator

To use the nPr calculator:

1. Enter the total number of items (n).

2. Enter the number of items to choose (r).

3. Toggle repetition on or off, and pick your output mode.

4. Review the number of permutations (nPr) and the step-by-step work shown below.

Example input: Total Items (n) = 9, Items to Choose (r) = 3.

Output:

Permutations (9P3) = 504
Combinations (9C3) = 84
Step-by-step work: 9P3 = 9!/(9-3)! = 9!/6! = 9×8×7 = 504

Another example: Total Items (n) = 5, Items to Choose (r) = 2.

Permutations (5P2) = 20
Combinations (5C2) = 10
Step-by-step work: 5P2 = 5!/(5-2)! = 5!/3! = 5×4 = 20

What is a Permutation?

A permutation is the number of ways to arrange a set of items where the order of the items matters.

Think of it like this: if you're lining people up for a photo, the order in which they stand matters. Alice first and Bob second is different from Bob first and Alice second.

Key characteristics:

Items are distinct (different from each other)
Order matters (arrangements)
No repetition allowed (usually — toggle repetition on for cases where it is)

Simple example: How many ways to arrange 2 letters from {A, B, C}?

Permutations: AB, AC, BA, BC, CA, CB = 6 ways
Notice: AB is different from BA!

The permutation formula:

nPr = n! / (n - r)!
Where n = total items, r = items to choose

Permutation vs Combination — What's the Difference?

Feature comparison:

Order: Permutations — order matters. Combinations — order doesn't matter.
Also called: Permutations — arrangements. Combinations — selections.
Example: Permutations — AB ≠ BA. Combinations — AB = BA.
Formula: Permutations — n! / (n-r)!. Combinations — n! / (r!(n-r)!).
Value: Permutations are always larger. Combinations are always smaller.
Used for: Permutations — passwords, seating, rankings. Combinations — lottery, teams, sampling.

Example: Set {A, B, C}, choose 2.

Permutations (order matters): AB, AC, BA, BC, CA, CB = 6
Combinations (order doesn't matter): AB, AC, BC = 3

When to use which:

Passwords → Permutations. Order of characters matters.
Lottery → Combinations. Order of numbers doesn't matter.
Seating arrangements → Permutations. Where people sit matters.
Choosing a team → Combinations. Who's on the team, not their order.
Race rankings → Permutations. First, second, third matters.
Picking flavors → Combinations. Order of flavors doesn't matter.

The Permutation Formula (nPr)

The permutation formula calculates the number of ways to arrange r items from n distinct items.

Formula:

nPr = n! / (n - r)!

Where:

n = total number of items
r = number of items to arrange
! = factorial (multiply all numbers from 1 to that number)

Expanded form:

nPr = n × (n-1) × (n-2) × ... × (n - r + 1)

Example: 9P3 = 9 × 8 × 7 = 504 (you multiply 3 numbers starting from 9).

Special cases:

nP0 = 1 (choose nothing)
nP1 = n (choose one item)
nPn = n! (arrange all items)
0! = 1 (mathematical convention)

How to Calculate Permutations — Step by Step

Calculating permutations is simple when you follow these steps.

Step 1: Identify n (total items) and r (items to choose). Example: 9 items, choose 3 → n = 9, r = 3.

Step 2: Write the formula: nPr = n! / (n-r)!

Step 3: Plug in the numbers: 9P3 = 9! / (9-3)! = 9! / 6!

Step 4: Expand the factorial (or multiply): 9! / 6! = 9 × 8 × 7 × 6! / 6! — the 6! cancels out — 9 × 8 × 7 = 504.

Answer: 9P3 = 504

More examples:

5P2 = 5!/(5-2)! = 5×4 = 20
6P3 = 6!/(6-3)! = 6×5×4 = 120
10P4 = 10!/(10-4)! = 10×9×8×7 = 5040
7P2 = 7!/(7-2)! = 7×6 = 42
8P4 = 8!/(8-4)! = 8×7×6×5 = 1680

How to Calculate Combinations (nCr)

A combination is the number of ways to choose r items from n items where order does not matter.

Formula:

nCr = n! / (r!(n - r)!)

Example:

9C3 = 9! / (3!(9-3)!) = 9! / (3! × 6!)
9C3 = (9 × 8 × 7) / (3 × 2 × 1)
9C3 = 504 / 6 = 84

Relationship between permutations and combinations: nCr = nPr / r!

Example: 9P3 = 504, 9C3 = 504 / 6 = 84 ✅

More examples:

5C2 = 5!/(2!×3!) = (5×4)/(2×1) = 10
6C3 = 6!/(3!×3!) = (6×5×4)/(3×2×1) = 20
10C4 = 10!/(4!×6!) = (10×9×8×7)/(4×3×2×1) = 210
7C2 = 7!/(2!×5!) = (7×6)/(2×1) = 21

Real-World Applications of Permutations and Combinations

2. Passwords (Permutations). Problem: how many 4-digit passwords using digits 0-9 without repetition? Solution: 10P4 = 10 × 9 × 8 × 7 = 5040. Why: order of digits matters for passwords.

3. Lottery (Combinations). Problem: pick 6 numbers from 49 in a lottery. Solution: 49C6 = 13,983,816. Why: order doesn't matter — {1,2,3,4,5,6} is the same as {6,5,4,3,2,1}.

4. Card Games (Combinations). Problem: how many 5-card poker hands from a 52-card deck? Solution: 52C5 = 2,598,960. Why: order of cards in your hand doesn't matter.

5. Race Rankings (Permutations). Problem: 8 runners — how many ways to award gold, silver, bronze? Solution: 8P3 = 8 × 7 × 6 = 336. Why: order matters — first, second, third are different.

6. Choosing a Team (Combinations). Problem: choose 3 people from 10 for a committee. Solution: 10C3 = 120. Why: the order on the committee doesn't matter.

Practice Problems

Problem 1: How many ways can you arrange 3 letters from the word "MATH"? Solution: n = 4 (M, A, T, H), r = 3 → 4P3 = 4 × 3 × 2 = 24.

Problem 2: How many ways can you choose 2 flavors from 5 ice cream flavors? Solution: n = 5, r = 2 → 5C2 = (5 × 4) / (2 × 1) = 10.

Problem 3: A teacher needs to assign 3 different roles from 8 students. How many ways? Solution: n = 8, r = 3 → 8P3 = 8 × 7 × 6 = 336.

Problem 4: How many different 4-digit PINs can be made from digits 0-9 without repetition? Solution: n = 10, r = 4 → 10P4 = 10 × 9 × 8 × 7 = 5040.

Problem 5: A pizza shop offers 10 toppings. How many ways to choose 3 toppings? Solution: n = 10, r = 3 → 10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120.

Answers: 1) 24 — 2) 10 — 3) 336 — 4) 5040 — 5) 120

Common Permutation Mistakes

Confusing permutations and combinations. Mistake: using permutations when order doesn't matter. Fix: ask "Does order matter?" If no → use combinations.
Forgetting to subtract r from n. Mistake: nPr = n! / r! (wrong). Fix: nPr = n! / (n-r)!
Not simplifying the factorial. Mistake: 5P2 = 5! / 3! = 120 / 6 = 20 (correct but more work). Fix: 5P2 = 5 × 4 = 20 (easier!).
Misreading the problem. Mistake: using nCr when the problem asks for nPr. Fix: look for keywords like "arrange," "order," or "positions" → permutations.
Forgetting that 0! = 1. Mistake: 5P0 = 0 (wrong). Fix: 5P0 = 1 (choose nothing, 1 way).
Using wrong values. Mistake: using the wrong numbers for n or r. Fix: carefully read the problem to identify n and r.
Not considering repetition. Mistake: using the no-repetition formula for problems where repetition is allowed. Fix: toggle repetition on — the formula becomes n^r.
Not checking answer reasonableness. Mistake: getting an answer that seems too large or too small. Fix: does the answer make sense? If not, double-check.

What This Calculator Can and Cannot Do

This calculator CAN help with:

Calculating permutations (nPr) and combinations (nCr), with or without repetition
Circular (round-table) and linear seating permutations
Factorial calculations, including very large factorials shown in scientific notation
Step-by-step work shown for every calculation
Understanding order matters vs order doesn't matter
Real-world applications — seating, passwords, lottery, cards, races, committees, and locks — in the Scenarios tab
Both small and very large numbers, with results that stay accurate even past standard number limits

This calculator CANNOT do:

Handle permutations of sets with repeated, indistinguishable items (e.g. arranging the letters of "MISSISSIPPI") — that requires a multinomial formula
Calculate probabilities directly
Replace understanding of the underlying concepts

Final Summary

Remember the key formulas:

Permutations (nPr): n! / (n-r)! — order matters
Combinations (nCr): n! / (r!(n-r)!) — order doesn't matter
Relationship: nCr = nPr / r!

Quick reference:

5P2 = 20 (arrangements)
5C2 = 10 (selections)
10P3 = 720
10C3 = 120

Start calculating now! Just enter your values above and get instant results — completely free.

Related concepts

Factorial

The product of all positive integers up to n, written n!, and the building block of every permutation and combination formula.

Fundamental Counting Principle

If one event can happen in m ways and a second in n ways, both together can happen in m × n ways — the basis of combinatorics.

Combinatorics

The branch of mathematics concerned with counting, arranging, and combining objects, underlying probability and statistics.

Frequently Asked Questions

What is a permutation calculator?+

A permutation calculator finds the number of ways to arrange r items from n distinct items where order matters.

What is the permutation formula?+

The permutation formula is nPr = n! / (n-r)!.

What is the difference between permutation and combination?+

In permutations, order matters (arrangements). In combinations, order doesn't matter (selections).

When do I use permutations?+

Use permutations when the order of items matters, such as seating arrangements, passwords, or rankings.

When do I use combinations?+

Use combinations when the order of items doesn't matter, such as lottery numbers, choosing a team, or picking flavors.

What is nPr?+

nPr is the notation for permutations. It means the number of ways to arrange r items from n distinct items.

What is the relationship between permutations and combinations?+

nCr = nPr / r!. Combinations are smaller because order doesn't matter.

How do I compute permutations?+

To compute permutations: identify n and r, use the formula nPr = n!/(n-r)!, and calculate.

What is 0!?+

0! = 1 (mathematical convention).

Can I use permutations with repetition?+

The standard permutation formula (nPr) is for arrangements without repetition. This calculator also supports repetition — toggle "Allow repetition" and it switches to the n^r formula automatically.

What is the difference between nPr and nCr?+

nPr (permutations) is for arrangements where order matters. nCr (combinations) is for selections where order doesn't matter.

How many ways can I arrange 5 books on a shelf?+

5P5 = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways.

How many ways to choose 3 toppings from 10?+

10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120 ways.

What is a possible combinations calculator?+

A possible combinations calculator calculates the number of ways to choose r items from n items where order doesn't matter (combinations).

Is this permutation calculator free?+

Yes — completely free with no registration required.

Permutation Calculator — Calculate nPr & Combinations Instantly

Inputs

Results

Bookmark this permutation calculator

Permutation & Combination Formulas

How the calculation works

Combinatorics Example — Choosing and Arranging 3 Candidates from 9

Learn more about this calculation

Standard Deviation Guide: Measure Data Spread Clearly

Random Number Generation Guide: Seeded PRNGs, True Randomness, and Safe Use Cases

Combination Calculator

Factorial & Gamma Calculator

Random Number Calculator

Descriptive Statistics Calculator

Permutation Calculator — Calculate nPr & Combinations Instantly

Inputs

Results

Bookmark this permutation calculator

Key output metrics

Permutations vs Combinations Growth

How to understand and use this calculator

Quick Answer

Use the Permutation Calculator

What is a Permutation?

Permutation vs Combination — What's the Difference?

The Permutation Formula (nPr)

How to Calculate Permutations — Step by Step

How to Calculate Combinations (nCr)

Real-World Applications of Permutations and Combinations

Practice Problems

Common Permutation Mistakes

What This Calculator Can and Cannot Do

Final Summary

Related concepts

Frequently Asked Questions

Permutation & Combination Formulas

How the calculation works

Combinatorics Example — Choosing and Arranging 3 Candidates from 9

Learn more about this calculation

Standard Deviation Guide: Measure Data Spread Clearly

Random Number Generation Guide: Seeded PRNGs, True Randomness, and Safe Use Cases

Combination Calculator

Factorial & Gamma Calculator

Random Number Calculator

Descriptive Statistics Calculator

Key output metrics

Permutations vs Combinations Growth

How to understand and use this calculator

Quick Answer

Use the Permutation Calculator

What is a Permutation?

Permutation vs Combination — What's the Difference?

The Permutation Formula (nPr)

How to Calculate Permutations — Step by Step

How to Calculate Combinations (nCr)

Real-World Applications of Permutations and Combinations

Practice Problems

Common Permutation Mistakes

What This Calculator Can and Cannot Do

Final Summary

Related concepts

Frequently Asked Questions