1. What Are Combinations and Permutations?

Combinations and permutations are fundamental concepts in combinatorics that count the number of ways to select items from a larger set. Combinations (C(n,r)) count selections where order doesn't matter, while permutations (P(n,r)) count ordered arrangements.

2. How the Calculator Works

The calculator uses these mathematical formulas:

Where:

• \( n! \) — Factorial of n (n × (n-1) × ... × 1)
• \( r! \) — Factorial of r
• \( (n-r)! \) — Factorial of (n minus r)

Explanation: The combination formula divides by r! to account for unordered selections, while the permutation formula maintains the order of selection.

3. Practical Applications

Details: These calculations are essential in probability, statistics, game design, cryptography, and any field requiring counting of possible arrangements or selections.

4. Using the Calculator

Tips: Enter positive integers where n ≥ r ≥ 0. The calculator will compute both combinations and permutations simultaneously.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combinations and permutations?
A: Combinations are for unordered selections (like lottery numbers), while permutations are for ordered arrangements (like passwords).

Q2: What if r > n?
A: By definition, both C(n,r) and P(n,r) are 0 when r > n, as you can't select more items than you have.

Q3: How does the calculator handle large numbers?
A: It uses factorial calculations which grow extremely rapidly. Very large inputs may exceed PHP's number handling capacity.

Q4: Are there real-world examples of these calculations?
A: Yes! Combinations help calculate lottery odds, while permutations are used in password strength calculations.

Q5: What about combinations with repetition?
A: This calculator uses the standard formulas without repetition. Different formulas apply when items can be selected multiple times.