1. What is the Combinations Formula?

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

2. How Does the Calculator Work?

The calculator uses the combinations formula:

Where:

• \( n! \) — Factorial of n (n × (n-1) × ... × 1)
• \( k! \) — Factorial of k
• \( (n - k)! \) — Factorial of (n - k)

Explanation: The formula divides the total permutations by the number of redundant arrangements (since order doesn't matter in combinations).

3. Importance of Combinations Calculation

Details: Combinations are essential for calculating probabilities, analyzing possible outcomes, and solving problems in fields like genetics, game theory, and experimental design.

4. Using the Calculator

Tips: Enter the total number of items (n) and the number to choose (k). Both must be non-negative integers with k ≤ n. The calculator handles values up to n=170 (limited by floating-point precision).

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. ABC and BAC are the same combination but different permutations.

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: What are some practical applications?
A: Lottery odds calculation, committee formation, card game probabilities, and genetic inheritance patterns.

Q4: Why does 0! equal 1?
A: This is a mathematical convention that makes the formula work correctly when k=0 or k=n.

Q5: How does this relate to Pascal's Triangle?
A: Each entry in Pascal's Triangle corresponds to C(n,k) for row n and position k.