1. What Are Combinations?

Combinations refer to the selection of items from a larger set where order doesn't matter. In mathematics, combinations are different from permutations, where order is important.

2. How Combinations Are Calculated

Combinations are calculated using the formula:

Where:

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

Explanation: The formula counts all possible ways to choose r items from n items without considering the order of selection.

3. Practical Applications

Details: Combinations are used in probability, statistics, game design, cryptography, and anywhere you need to count possibilities without regard to order.

4. Using the Calculator

Tips: Enter the total number of items (n) to calculate all possible combinations for every possible selection size (r) from 0 to n.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combinations and permutations?
A: Combinations don't consider order (AB = BA), while permutations do (AB ≠ BA).

Q2: Why calculate C(n,0) and C(n,n)?
A: C(n,0) = 1 (one way to choose nothing) and C(n,n) = 1 (one way to choose everything).

Q3: What's the largest n this calculator can handle?
A: The calculator is limited to n ≤ 100 to prevent excessive computation.

Q4: How are combinations related to Pascal's Triangle?
A: Each entry in Pascal's Triangle corresponds to a combination number C(n,r).

Q5: What if I need combinations with repetition?
A: This calculator computes combinations without repetition. The formula changes to C(n+r-1,r) when repetition is allowed.