1. What is the Combination Formula?

The combination formula calculates the number of ways to choose k items from n items without regard to order. It's fundamental in probability, statistics, and combinatorics.

2. How Does the Calculator Work?

The calculator uses the combination formula:

Where:

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

Explanation: The formula divides the total permutations by the number of redundant arrangements to get unique combinations.

3. Importance of Combinations

Details: Combinations are essential for calculating probabilities, analyzing possible outcomes, and solving problems in fields from genetics to computer science.

4. Using the Calculator

Tips: Enter positive integers where n ≥ k ≥ 0. The calculator computes exact integer results for valid inputs.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combinations and permutations?
A: Combinations consider only selection, while permutations consider both selection and arrangement order.

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

Q3: How does this relate to binomial coefficients?
A: Combination values are exactly the binomial coefficients seen in algebra's binomial theorem.

Q4: What's the largest n this calculator can handle?
A: Due to factorial growth, accurate results are limited to n ≤ 170 (floating-point precision limit).

Q5: Are there real-world applications?
A: Yes! From lottery odds calculation to experimental design in research and password complexity analysis.