Permutation and Combination Calculator
Calculate permutations and combinations for probability, statistics, and combinatorial problems. Perfect for students, researchers, and professionals working with discrete mathematics.
Calculator Input
Results
Step-by-Step Calculation
How to Use
Basic Steps
- Select the calculation type (permutation, combination, or both)
- Enter the total number of items (n)
- Enter the number of items to select (r) if applicable
- Choose whether repetition is allowed
- View the calculated results instantly
Input Guidelines
- n must be a non-negative integer (0 ≤ n ≤ 170)
- r must be less than or equal to n
- Use quick examples for common scenarios
- Large numbers may be displayed in scientific notation
Understanding Permutations and Combinations
Permutations
Definition: The number of ways to arrange r items from n total items where order matters.
Formula:
With repetition: P(n,r) = n^r
Example: Arranging 3 people from 5 candidates for president, vice-president, and secretary positions.
Combinations
Definition: The number of ways to select r items from n total items where order doesn't matter.
Formula:
With repetition: C(n,r) = (n+r-1)! / (r! × (n-1)!)
Example: Selecting 3 people from 5 candidates for a committee where roles are equal.
Key Concepts
Factorial (n!)
The product of all positive integers from 1 to n. Used as the foundation for permutation and combination calculations.
Order Matters
Permutations consider order (ABC ≠ BAC), while combinations don't (ABC = BAC = CAB).
Repetition
With repetition allows selecting the same item multiple times. Without repetition means each item can only be selected once.
Example Calculations
Example 1: Lottery Numbers
Problem: How many ways can you choose 6 numbers from 49 for a lottery?
Solution: This is a combination problem (order doesn't matter)
Answer: There are 13,983,816 possible combinations.
Example 2: Race Positions
Problem: In how many ways can 8 runners finish 1st, 2nd, and 3rd?
Solution: This is a permutation problem (order matters)
Answer: There are 336 possible arrangements.
Frequently Asked Questions
When should I use permutations vs combinations?
Use permutations when the order of selection matters (like race positions, passwords, or seating arrangements). Use combinations when order doesn't matter (like selecting team members, lottery numbers, or choosing items from a menu).
What's the difference between with and without repetition?
Without repetition means each item can only be selected once (like choosing unique people for a committee). With repetition allows selecting the same item multiple times (like rolling dice or drawing cards with replacement).
Why are there limits on the input values?
Factorial calculations grow extremely quickly. Values above 170 would result in numbers too large for standard computation. For practical problems, these limits are usually sufficient.