1. What is Combination?

A combination is a selection of items from a larger set where the order of selection does not matter. It's a fundamental concept in combinatorics and probability.

2. How Does the Calculator Work?

The calculator uses the combination formula:

Where:

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

Explanation: The formula calculates how many ways you can choose r items from n items without considering the order of selection.

3. Importance of Combination Calculation

Details: Combinations are essential in probability theory, statistics, and many real-world applications like lottery calculations, team selections, and password combinations.

4. Using the Calculator

Tips: Enter the total number of items (n) and the number to choose (r). Both must be non-negative integers with r ≤ n.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combination and permutation?
A: Combinations consider only selection, while permutations consider both selection and order. ABC is the same combination as BAC but different permutations.

Q2: What if r > n?
A: By definition, C(n, r) = 0 when r > n since you can't choose more items than you have.

Q3: What are some real-world applications?
A: Lottery odds calculation, forming committees, creating unique test groups, and cryptography.

Q4: How does this relate to Pascal's Triangle?
A: Each entry in Pascal's Triangle corresponds to C(n, r) where n is the row number and r is the position in the row.

Q5: What's the largest n this calculator can handle?
A: Due to factorial growth, values above n=170 may cause overflow issues in standard calculations.