Permutation & Combination Calculator
Calculate arrangements (nPr) and selections (nCr) with ease
How to Use This Permutation & Combination Calculator
- Enter the total number of items (n) - this is your complete set
- Enter the number of items to select (r) - how many you're choosing
- Click Calculate to see both permutation (nPr) and combination (nCr) results
- Review the factorial breakdown and step-by-step calculations
Example: Selecting 3 winners from 10 contestants: If the positions matter (1st, 2nd, 3rd place), use permutation: 10P3 = 720 ways. If you're just picking 3 people for a committee, use combination: 10C3 = 120 ways.
Tip: Ask yourself: 'Does order matter?' If swapping items creates a different outcome, use permutation. If not, use combination.
Why Use a Permutation & Combination Calculator?
Permutations and combinations solve counting problems in probability, statistics, and real-world planning scenarios.
- Calculate lottery odds by finding how many number combinations are possible
- Plan tournament brackets and determine possible seating arrangements
- Figure out how many different pizza topping combinations you can create
- Determine password security by counting possible character permutations
- Solve probability homework problems involving selection and arrangement
- Plan team selections, committee formations, and group assignments
Understanding Your Results
Permutations always produce larger numbers than combinations for the same n and r values.
| Result | Meaning | Action |
|---|---|---|
| nPr (Permutation) | Order matters | Use for rankings, sequences, passwords, race positions |
| nCr (Combination) | Order doesn't matter | Use for teams, groups, lottery picks, card hands |
| nPr = nCr x r! | Relationship between them | Permutations = combinations multiplied by arrangements of r items |
Meaning: Order matters
Action: Use for rankings, sequences, passwords, race positions
Meaning: Order doesn't matter
Action: Use for teams, groups, lottery picks, card hands
Meaning: Relationship between them
Action: Permutations = combinations multiplied by arrangements of r items
Note: When r = 1, both values equal n. When r = n, permutation = n! and combination = 1.
About Permutation & Combination Calculator
Formula
nPr = n!/(n-r)! | nCr = n!/(r!(n-r)!) For permutations, divide n! by (n-r)! to get ordered arrangements. For combinations, also divide by r! to remove duplicate orderings of the same selection.
Current Standards: Maximum n = 170 due to JavaScript number limits. Results above 10^15 display in scientific notation.
Frequently Asked Questions
When should I use permutation vs combination?
Use permutation when order creates different outcomes - like arranging books on a shelf (ABC differs from BAC). Use combination when order doesn't matter - like choosing 3 toppings for pizza (pepperoni-mushroom-olive is the same as olive-mushroom-pepperoni). The key question: does rearranging the same items count as different?
Why is the maximum n value limited to 170?
Factorials grow extremely fast - 170! is approximately 7.26 x 10^306, which is near JavaScript's maximum safe number. Beyond 170, calculations overflow and produce Infinity. For larger values, you'd need specialized big-number libraries.
How do I calculate lottery odds?
Lotteries typically use combinations since draw order doesn't matter. For a 6/49 lottery (pick 6 from 49 numbers), calculate 49C6 = 13,983,816 possible combinations. Your odds of winning the jackpot are 1 in approximately 14 million.
What does r! represent in the combination formula?
r! accounts for the duplicate arrangements. When selecting 3 items, those same 3 items can be arranged in 3! = 6 different orders. Since combinations treat all these arrangements as one selection, we divide by r! to eliminate overcounting.
Can r be larger than n?
No - you can't select more items than exist in the set. If r > n, both permutation and combination are mathematically undefined (you'd be dividing by a negative factorial). The calculator validates this and shows an error.