🔢 Factorial & P/C Calculator

Calculate factorials, permutations, and combinations

Factorial Calculation

n! = n × (n-1) × (n-2) × ... × 2 × 1
Factorial of n is the product of all positive integers ≤ n

Common Examples

5! (Factorial)
Calculate 5 factorial = 120
P(10,3) - Arranging 3 from 10
Race positions: 1st, 2nd, 3rd from 10 runners
C(52,5) - Poker Hand
Choose 5 cards from 52-card deck
C(49,6) - Lottery
Choose 6 numbers from 49
P(26,4) - Password
4-letter password from alphabet

🔢 Complete Guide to Factorial, Permutation & Combination Calculator

Master counting principles and combinatorics

A factorial, permutation, and combination calculator is an essential tool for solving counting problems in mathematics, statistics, and probability. Whether you're working on combinatorics homework, calculating lottery odds, or analyzing arrangements, our online calculator provides instant, accurate results for all three operations right in your browser.

🎯 What is This Calculator?

This calculator performs three fundamental counting operations: Factorial (n!) calculates the product of all positive integers up to n, Permutation (nPr) calculates ordered arrangements of r items from n items, and Combination (nCr) calculates unordered selections of r items from n items. These are essential for probability, statistics, and combinatorics.

💡 Why Use an Online Calculator?

Our online calculator offers several advantages: instant calculations for factorial, permutation, and combination, handles large numbers, shows formulas and step-by-step solutions, works on all devices, and requires no installation. Perfect for students, statisticians, and anyone working with counting problems!

🏃 Common Uses

📚
Math Homework
Solve combinatorics problems. Calculate factorials, permutations, combinations.
🎲
Probability
Calculate odds and probabilities. Determine possible outcomes.
🎰
Lottery & Games
Calculate lottery odds. Determine winning probabilities.
📊
Statistics
Sampling methods, experimental design, data analysis.
🔬
Science
Experimental arrangements, molecular combinations, genetics.
💼
Business
Team arrangements, scheduling, resource allocation.

📖 How to Use Our Calculator

Using our calculator is simple:

  1. Choose Operation: Select Factorial, Permutation, or Combination tab
  2. Enter Values: Input n (and r for permutation/combination)
  3. Calculate: Click to see results
  4. View Results: See the answer with formula
  5. Try Examples: Use quick example buttons to learn
📝 Example: Combination

Problem: Choose 3 people from 10 (order doesn't matter)

Solution: C(10,3) = 10!/(3!×7!) = 120 ways

⚡ Pro Tips

1
Order Matters?
Use Permutation if order matters (ABC ≠ BAC). Use Combination if order doesn't matter (ABC = BAC).
2
Factorial Growth
Factorials grow extremely fast! 10! = 3,628,800 and 20! = 2.4 quintillion.
3
nPr vs nCr
nPr ≥ nCr always. Permutations count more arrangements because order matters.
4
Special Cases
0! = 1 (by definition). C(n,0) = 1 and C(n,n) = 1 (only one way to choose none or all).

🔑 Key Features

  • Three Operations: Factorial, permutation, and combination in one tool.
  • Large Numbers: Handles calculations up to reasonable limits.
  • Formulas Shown: See the mathematical formulas used.
  • Instant Results: Calculate immediately.
  • Quick Examples: One-click example problems.
  • Tabbed Interface: Easy switching between operations.
  • Mobile Friendly: Works perfectly on all devices.
  • No Installation: Works directly in your browser.
  • Always Free: No subscriptions or hidden fees.
  • Accurate: Precise mathematical calculations.

💪 Benefits

Having a reliable calculator offers numerous advantages:

🎯 Solve Counting Problems Fast

Eliminate manual calculations and avoid errors. Perfect for homework, probability calculations, and understanding combinatorics concepts.

  • Save Time: Instant calculations instead of manual work.
  • Avoid Errors: Accurate results every time.
  • Learn Concepts: See formulas and understand how they work.
  • Verify Work: Check your manual calculations.
  • Handle Large Numbers: Calculate factorials beyond manual capability.
  • Always Available: Access from any device.

🎓 Understanding the Concepts

Factorial (n!)

Formula: n! = n × (n-1) × (n-2) × ... × 2 × 1

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

Use: Total arrangements of n distinct items

Permutation (nPr)

Formula: P(n,r) = n!/(n-r)!

Example: P(5,3) = 5!/(5-3)! = 120/2 = 60

Use: Ordered arrangements of r items from n items (order matters)

Combination (nCr)

Formula: C(n,r) = n!/(r!×(n-r)!)

Example: C(5,3) = 5!/(3!×2!) = 120/(6×2) = 10

Use: Unordered selections of r items from n items (order doesn't matter)

When to Use Each

  • Factorial: Arranging all items (seating n people, ordering n books)
  • Permutation: Arranging some items where order matters (race podium, passwords)
  • Combination: Selecting items where order doesn't matter (lottery, committees)

Real-World Examples

  • Factorial: 5 people in a row = 5! = 120 ways
  • Permutation: Top 3 from 10 runners = P(10,3) = 720 ways
  • Combination: Choose 6 from 49 lottery numbers = C(49,6) = 13,983,816 ways
🎲 Real-World: Lottery Odds

Problem: What are the odds of winning a 6/49 lottery?

Solution: C(49,6) = 13,983,816 possible combinations. Odds = 1 in 13,983,816!

❓ Frequently Asked Questions (FAQ)

What is a factorial?
A factorial (n!) is the product of all positive integers from 1 to n. For example, 5! = 5×4×3×2×1 = 120. It represents total arrangements of n items.
What's the difference between permutation and combination?
Permutation counts ordered arrangements (ABC ≠ BAC). Combination counts unordered selections (ABC = BAC). Use permutation when order matters, combination when it doesn't.
Why is 0! equal to 1?
0! = 1 by mathematical definition. This makes formulas work correctly and represents that there's exactly one way to arrange zero items (do nothing).
When do I use permutation vs combination?
Use permutation when order matters (race positions, passwords, seating arrangements). Use combination when order doesn't matter (lottery numbers, committees, card hands).
How do I calculate lottery odds?
Use combinations! For a 6/49 lottery, calculate C(49,6) to find total possible combinations. Your odds of winning are 1 in C(49,6) = 1 in 13,983,816.
What's the maximum n I can calculate?
Our calculator handles reasonable values. Very large factorials (n > 170) may overflow. For permutations and combinations, we can handle larger values.
Can r be greater than n?
No! You can't select or arrange more items (r) than you have available (n). Always ensure r ≤ n for valid results.
Will this work on my phone?
Yes! The calculator is fully responsive and works perfectly on smartphones and tablets. Calculate factorials, permutations, and combinations anywhere.
Can I use this for homework?
Absolutely! Our calculator is perfect for checking homework answers, learning combinatorics concepts, and understanding counting principles.
How accurate is this calculator?
Our calculator uses standard mathematical formulas and is completely accurate within computational limits. Results are precise for academic and professional use.

🚀 Getting Started

Ready to calculate? Scroll back up and choose your operation! Select Factorial, Permutation, or Combination, enter your values, and see instant results with formulas. Try the example buttons to learn different counting scenarios!

🎯 Quick Start Tips

  • Choose Factorial for arranging all n items
  • Choose Permutation when order matters (nPr)
  • Choose Combination when order doesn't matter (nCr)
  • Try example buttons to see how each works
  • Remember: nPr ≥ nCr always!

📊 Conclusion

A factorial, permutation, and combination calculator is an indispensable tool for students, statisticians, and anyone working with counting problems. Our free online calculator provides instant, accurate calculations for all three fundamental counting operations.

Whether you're solving combinatorics homework, calculating lottery odds, analyzing arrangements, or working on probability problems, our calculator gives you the accurate results you need with an easy-to-use interface and clear formulas.

Start calculating now and master counting principles with confidence!