Advertisement Space
Permutation & Combination Calculator
Results
-
Permutation & Combination Formulas
Permutation Formula (nPr)
Number of ways to arrange r items from n items where order matters:
This is used when the arrangement/order is important.
Combination Formula (nCr)
Number of ways to choose r items from n items where order doesn't matter:
This is used when only the selection matters, not the order.
Factorial Formula (n!)
Product of all positive integers from 1 to n:
Note: 0! = 1 (by definition)
Key Terms
- n = Total number of items available
- r = Number of items to select/arrange
- n! = Factorial of n (n × (n-1) × (n-2) × ... × 1)
- nPr = Permutation (order matters)
- nCr = Combination (order doesn't matter)
Permutation vs Combination
| Feature | Permutation (nPr) | Combination (nCr) |
|---|---|---|
| Definition | Arrangements of items where order matters | Selection of items where order doesn't matter |
| Formula | nPr = n! / (n-r)! | nCr = n! / (r! × (n-r)!) |
| Result Size | Generally larger values | Generally smaller values |
| Use Cases | Passwords, rankings, seating arrangements | Teams, committees, lottery numbers |
| Example | ABC, ACB, BAC are different | ABC, ACB, BAC are the same |
How to Use the Calculator
Step 1: Choose Your Type
Select between Permutation (nPr) or Combination (nCr) using the tabs based on whether order matters in your problem.
Step 2: Enter Values
Input n (total items) and r (items to select/arrange). Remember that r cannot be greater than n.
Step 3: Calculate
Click the "Calculate" button to see the result, formula used, and detailed calculation breakdown.
Choosing Between Permutation and Combination
- Use Permutation (nPr) if: Order matters, you're arranging items, creating passwords, rankings, or sequences
- Use Combination (nCr) if: Order doesn't matter, you're selecting a team, committee, choosing lottery numbers
Real-World Applications
Password & Security
Creating secure passwords uses permutations because the order of characters matters. P(26,8) gives possible 8-character passwords from 26 letters.
Sports & Rankings
Determining podium finishes (1st, 2nd, 3rd) uses permutations since finishing position matters. P(10,3) = 720 ways for 10 runners to finish in top 3.
Committee Formation
Selecting committee members uses combinations since roles are equal. C(15,5) = 3003 ways to choose 5 people from 15 for a committee.
Lottery & Gambling
Lottery numbers use combinations (order doesn't matter). C(49,6) = 10,068,347 possible combinations for a 6/49 lottery.
Business & Sales
Sales teams arranging client visits (order matters) = Permutation. Selecting products for a bundle (order doesn't matter) = Combination.
Medicine & Genetics
Genetic combinations determine traits. C(4,2) = 6 ways blood type can combine from 4 alleles.
Education & Testing
Multiple choice question arrangements use permutations. Answer key sequences must be in specific order.
Event Planning
Seating arrangements for a dinner = Permutation. Selecting guests from an invite list = Combination.
Common Examples
Example 1: Arranging Books (Permutation)
How many ways can you arrange 3 books from a shelf of 5 books?
Answer: P(5,3) = 5! / 2! = 120 / 2 = 60 ways
Because which book comes first, second, and third matters.
Example 2: Selecting Team Members (Combination)
How many ways can you select 2 players from a team of 5?
Answer: C(5,2) = 5! / (2! × 3!) = 120 / (2 × 6) = 10 ways
Because it doesn't matter who is selected first or second, just which 2 are on the team.
Example 3: PIN Codes (Permutation)
How many 4-digit PIN codes can be created from 10 digits (0-9)?
Answer: P(10,4) = 10! / 6! = 5,040 possible PIN codes
Because 1234 is different from 4321.
Example 4: Lottery Numbers (Combination)
How many ways to choose 6 numbers from 49?
Answer: C(49,6) = 10,068,347 possible combinations
Because 1-2-3-4-5-6 is the same as 6-5-4-3-2-1.
Frequently Asked Questions
What is factorial (n!)?
Factorial is the product of all positive integers up to n. For example: 5! = 5×4×3×2×1 = 120. By definition, 0! = 1.
Why is nPr always ≥ nCr?
Because permutation counts order, while combination doesn't. Multiple arrangements count as 1 combination. The relationship is nPr = nCr × r!
What if r > n?
If r > n, the result is undefined. You cannot select or arrange more items than you have available.
Can r equal 0?
Yes. P(n,0) = 1 and C(n,0) = 1, meaning there's exactly 1 way to arrange or choose 0 items (doing nothing).
How do I know which to use?
Ask yourself: Does the order matter? If YES = Permutation. If NO = Combination. Example: "First, second, third place" = order matters = Permutation.
What's the difference between arrangement and selection?
Arrangement (Permutation) cares about which position each item is in. Selection (Combination) only cares which items are chosen, not their positions.
Related Calculators
Explore our other mathematical and statistical calculators: