Permutation and Combination Calculator - Calculate P&C
Calculate permutations and combinations instantly with step-by-step factorial calculations and detailed explanations
Permutation & Combination Calculator
Results
What is a Permutation and Combination Calculator?
A Permutation and Combination Calculator is a free mathematical tool that computes P(n,r) and C(n,r) values instantly. It calculates how many ways you can arrange or select items from a set, using the November 2025 standard factorial formulas for precise results.
This calculator is essential for:
- Students and Teachers - Solve probability and statistics homework problems quickly
- Data Scientists - Calculate sample spaces and event probabilities
- Competitive Programmers - Compute combinatorial values for algorithm design
- Researchers - Analyze experimental designs and statistical models
To calculate probability values for statistical events, explore our Probability Calculator for computing event likelihoods and odds.
For binomial probability distributions with multiple trials, check out our Binomial Distribution Calculator to analyze success probabilities.
To perform advanced mathematical operations beyond P&C, use our Scientific Calculator for comprehensive computational functions.
How Permutation and Combination Formulas Work
The calculator uses November 2025 standard formulas with optimized factorial computation:
Arranges r items from n total items where ABC ≠ BAC
Selects r items from n total items where ABC = BAC
Special case: 0! = 1 by definition
Avoids full factorial to prevent overflow for large values
Example: For P(10,3): Instead of computing 10!/7!, we calculate 10×9×8 = 720. For C(10,3): We divide 720 by 3! = 6 to get 120 combinations.
Key Permutation & Combination Concepts
Permutation (P)
Arrangement of items where order matters. Example: Race positions - 1st, 2nd, 3rd are all different outcomes.
Combination (C)
Selection of items where order doesn't matter. Example: Team selection - choosing ABC is same as BAC.
Factorial (n!)
Product of all positive integers up to n. Critical for both formulas. 5! = 5×4×3×2×1 = 120.
Relationship
C(n,r) = P(n,r) / r! because combinations don't count different arrangements of same items.
How to Use This Calculator
Enter n Value
Input total number of items available (0-170)
Enter r Value
Input number of items to choose (r ≤ n)
Click Calculate
Get instant P(n,r) and C(n,r) results
View Factorials
See n!, r!, and (n-r)! calculations
Benefits of Using This Calculator
- • Instant Results: Calculate P(n,r) and C(n,r) simultaneously with one click for fast problem solving.
- • Handles Large Numbers: Efficiently computes factorials up to 170! using optimized algorithms to prevent overflow.
- • Step-by-Step Factorials: Shows n!, r!, and (n-r)! values for complete understanding of calculations.
- • Educational Explanations: Provides clear explanations of when to use permutations vs combinations.
- • Error-Free Accuracy: Eliminates manual calculation mistakes in complex factorial operations.
- • November 2025 Standard: Uses latest mathematical standards for permutation and combination formulas.
Important Considerations
1. Order Matters? Use Permutation
If the sequence is important (e.g., passwords, race rankings, seating arrangements), use P(n,r). ABC and BAC are different.
2. Order Doesn't Matter? Use Combination
If only selection matters (e.g., lottery numbers, team members, card hands), use C(n,r). ABC and BAC are the same.
3. r Must Be ≤ n
You cannot choose more items than available. If r > n, the calculator will show an error message.
4. Large Factorials
Factorials grow extremely fast. 20! = 2.4×10¹⁸. The calculator handles values up to 170! efficiently.
5. Special Cases
P(n,0) = C(n,0) = 1 (choosing nothing), P(n,n) = n! (full arrangement), C(n,n) = 1 (choosing all).
Frequently Asked Questions (FAQ)
Q: What is the difference between permutation and combination?
A: Permutation (P) considers order - selecting and arranging items where ABC is different from BAC. Combination (C) ignores order - selecting items where ABC is the same as BAC. Use permutations when order matters, combinations when it doesn't.
Q: What are the formulas for permutation and combination?
A: Permutation formula: P(n,r) = n! / (n-r)! where you arrange r items from n total items. Combination formula: C(n,r) = n! / (r!(n-r)!) where you select r items from n total items without regard to order.
Q: When should I use permutations vs combinations?
A: Use permutations for: arranging books on a shelf, race positions, password sequences, or any scenario where order matters. Use combinations for: selecting team members, choosing lottery numbers, forming committees, or any scenario where order doesn't matter.
Q: What is a factorial and how is it calculated?
A: A factorial (n!) is the product of all positive integers from 1 to n. For example: 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely fast - 10! = 3,628,800.