1. What is the Combination Formula?

The combination formula calculates the number of ways to choose r items from a set of n distinct items where order does not matter. It is represented as C(n, r) or "n choose r".

2. How Does the Calculator Work?

The calculator uses the combination formula:

Where:

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

Explanation: The formula calculates the number of unique subsets of size r that can be formed from a set of n distinct items, without considering the order of selection.

3. Importance of Combination Calculation

Details: Combination calculations are fundamental in probability theory, statistics, combinatorics, and various real-world applications such as lottery probabilities, team selections, and sampling methods.

4. Using the Calculator

Tips: Enter positive integer values for n and r, where n ≥ r ≥ 0. The calculator will compute the number of possible combinations.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combinations and permutations?
A: Combinations consider selection order irrelevant (ABC = ACB = BAC, etc.), while permutations treat different orders as distinct arrangements.

Q2: What if r > n?
A: The combination C(n, r) is defined as 0 when r > n, as you cannot choose more items than available.

Q3: What is the value of C(n, 0)?
A: C(n, 0) = 1, representing the single way to choose zero items from n items (the empty set).

Q4: Are there practical limits to the values of n and r?
A: Yes, factorial calculations grow extremely rapidly. For n > 170, factorial values exceed PHP's floating point precision limits.

Q5: Can this calculator handle non-integer values?
A: No, combination calculations require integer values for n and r as they represent counts of discrete items.