Combinatorics Calculator
Common Values
When to Use Each
Combination (Order doesn't matter)
Selecting a committee, lottery numbers, pizza toppings.
Permutation (Order matters)
Rankings, passwords, arranging people in line.
Factorial
Arranging all items: ways to arrange n distinct objects.
Quick Examples
Related Calculators
Understanding Combinatorics
What Is Combinatorics?
Combinatorics is the branch of mathematics dealing with counting, arrangements, and combinations of objects. It's fundamental to probability theory, statistics, and many real-world applications.
Key Concepts
Factorial (n!)
The product of all positive integers up to n.
Permutation P(n,r)
Arrangements where order matters.
Combination C(n,r)
Selections where order doesn't matter.
Permutation vs Combination
| Aspect | Permutation | Combination |
|---|---|---|
| Order | Matters | Doesn't matter |
| Formula | n!/(n-r)! | n!/(r!(n-r)!) |
| Example | PIN codes, rankings | Lottery, committees |
| P(5,3) vs C(5,3) | 60 | 10 |
Real-World Examples
Poker Hands (Combination)
How many 5-card hands from a 52-card deck?
C(52,5) = 52!/(5!×47!) = 2,598,960 possible hands
Race Results (Permutation)
Ways for 8 runners to finish 1st, 2nd, 3rd?
P(8,3) = 8!/(8-3)! = 8!/5! = 336 different outcomes
Seating Arrangement (Factorial)
Ways to arrange 6 people in 6 chairs?
6! = 6×5×4×3×2×1 = 720 arrangements
Important Properties
Useful Identities
Tips for Solving Problems
- Ask: "Does the order of selection matter?"
- If ABC ≠ BAC for your problem → Use Permutation
- If ABC = BAC for your problem → Use Combination
- For complex problems, break into smaller parts
- Check: Permutations are always ≥ Combinations for same n,r
Frequently Asked Questions
When should I use permutations vs combinations?
Use permutations when order matters (like arranging books on a shelf, race positions, or PIN codes). Use combinations when order doesn't matter (like choosing team members, lottery numbers, or committee selections). A simple test: if ABC and BAC are different outcomes in your problem, use permutations.
What does C(52,5) represent in poker?
C(52,5) = 2,598,960 represents the total number of possible 5-card poker hands from a standard 52-card deck. Since the order you receive cards doesn't matter (only which cards you have), we use combinations rather than permutations.
Why is 0! equal to 1?
0! = 1 by mathematical convention because it makes formulas work correctly. For example, C(n,n) should equal 1 (there's exactly one way to choose all items), and the formula n!/(n!*0!) only gives 1 if 0! = 1. It also represents the one way to arrange zero objects: do nothing.
How do I calculate nCr on a regular calculator?
Calculate n!/(r!(n-r)!) step by step. For C(10,3): First calculate 10! = 3,628,800, then 3! = 6, then 7! = 5,040. Finally: 3,628,800/(6 x 5,040) = 120. Alternatively, simplify first: C(10,3) = (10x9x8)/(3x2x1) = 720/6 = 120.