1. What Are Combinations Without Repetition?

Combinations without repetition refer to the number of ways to choose k items from a set of n distinct items where order doesn't matter and items cannot be repeated. This is also known as "n choose k" or binomial coefficients.

2. How The Formula Works

The combinations formula is:

Where:

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

Explanation: The formula divides the total permutations by k! to eliminate ordering, since combinations don't consider the order of selected items.

3. Practical Applications

Details: Combinations are used in probability theory, statistics, lottery calculations, team selection problems, and any scenario where you need to count possible selections without regard to order.

4. Using The Calculator

Tips: Enter the total number of items (n) and the number of items to choose (k). Both values must be non-negative integers, and k cannot exceed n.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combinations and permutations?
A: Combinations don't consider order (ABC = BCA), while permutations do (ABC ≠ BCA). Use combinations when order doesn't matter.

Q2: What if k = 0 or k = n?
A: C(n, 0) = 1 (one way to choose nothing) and C(n, n) = 1 (one way to choose everything).

Q3: Can combinations be used for repeated items?
A: No, this formula is specifically for combinations without repetition. For combinations with repetition, a different formula is used.

Q4: What are some real-world examples?
A: Selecting committee members from a group, choosing lottery numbers, forming teams from a pool of players.

Q5: How does this relate to binomial coefficients?
A: Combinations without repetition are exactly the binomial coefficients that appear in the binomial theorem expansion.