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 arrangements where order matters.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Where:

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

Explanation: The combination formula divides by r! to account for the different orderings that would count as the same combination. The permutation formula doesn't have this division because order matters.

3. Practical Applications

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

4. Using the Calculator

Tips: Enter the total number of items (n) and the number to select (r). Both must be non-negative integers with r ≤ n. The calculator will compute both combinations and permutations.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combinations and permutations?
A: Combinations are for selections where order doesn't matter (like lottery numbers). Permutations are for arrangements where order matters (like passwords).

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

Q3: How does this relate to the binomial coefficient?
A: The combination C(n,r) is exactly the binomial coefficient used in the binomial theorem.

Q4: What are some real-world examples?
A: Combinations: lottery tickets, committee selections. Permutations: password possibilities, race finishing orders.

Q5: How does this calculator handle large numbers?
A: It uses PHP's integer handling, but very large factorials (n > 20) may exceed integer limits and return inaccurate results.