1. What are Combinations and Permutations?

Combinations and permutations are mathematical concepts used to count arrangements of items. Permutations (P) consider order important, while combinations (C) do not.

2. How Does the Calculator Work?

The calculator uses the factorial formulas:

Where:

• \( n \) — Total number of items
• \( r \) — Number of items selected
• \( ! \) — Factorial (product of all positive integers ≤ the number)

Explanation: Permutations count ordered arrangements, while combinations count unordered groups.

3. Differences Between P and C

Key Difference: For the same n and r, P will always be larger than or equal to C because order matters in permutations. Example: ABC and BAC are different permutations but the same combination.

4. Using the Calculator

Tips: Enter positive integers where n ≥ r. The calculator will compute both permutation and combination counts. Values up to n=170 can be calculated accurately.

5. Frequently Asked Questions (FAQ)

Q1: When should I use permutation vs combination?
A: Use permutation when order matters (e.g., race rankings). Use combination when order doesn't matter (e.g., lottery numbers).

Q2: What if n = r?
A: P(n,n) = n! (all possible orderings). C(n,n) = 1 (only one way to choose all items).

Q3: What's the largest n this calculator can handle?
A: Up to n=170 before factorial values exceed floating point limits.

Q4: What about repetition?
A: This calculator assumes no repetition. Different formulas apply when items can be repeated.

Q5: Are there real-world applications?
A: Yes! Used in probability, statistics, cryptography, game theory, and anywhere counting arrangements is needed.