1. What Are Unique Combinations?

Combinations represent the number of ways to choose k items from a set of n items without regard to order. This is different from permutations where order matters.

2. How Does the Calculator Work?

The calculator uses the combinations formula:

Where:

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

Explanation: The formula divides the total permutations by the number of redundant arrangements (k! for the chosen items and (n-k)! for the unchosen items).

3. Importance of Combination Calculations

Details: Combination calculations are fundamental in probability, statistics, and many real-world applications like lottery odds, team selections, and experimental design.

4. Using the Calculator

Tips: Enter positive integers where n ≥ k ≥ 0. The calculator will compute the number of unique combinations possible.

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: 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's the value of C(n, 0)?
A: C(n, 0) = 1 for any n ≥ 0, representing the single way to choose nothing.

Q4: How does this relate to binomial coefficients?
A: Combination numbers are exactly the binomial coefficients seen in algebra's binomial theorem.

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