1. What is Combinations With Repetition?

Combinations with repetition (also called multichoose) calculates the number of ways to choose r items from n options where order doesn't matter and items can be chosen more than once. It's used in probability, statistics, and combinatorics.

2. How Does the Calculator Work?

The calculator uses the combinations with repetition formula:

Where:

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

Explanation: The numerator counts arrangements with repetition, while the denominator accounts for the indistinct order of selection.

3. Importance of Combinations Calculation

Details: This calculation is essential for problems involving selection with possible duplicates, such as counting possible outcomes in probability or distributing identical items.

4. Using the Calculator

Tips: Enter positive integers for both n (number of distinct items) and r (number to choose). The calculator will compute the number of possible combinations.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between combinations with and without repetition?
A: With repetition allows the same item to be chosen multiple times, while without repetition each item can be chosen only once.

Q2: What are some real-world applications?
A: Counting possible ice cream combinations (same flavor can be chosen multiple times), possible dice rolls, or ways to distribute identical items.

Q3: How does this relate to the "stars and bars" theorem?
A: The formula is mathematically equivalent to the stars and bars method of combinatorics.

Q4: What's the largest input this calculator can handle?
A: Due to factorial growth, inputs above ~20 may cause overflow. For larger numbers, logarithmic approaches or specialized libraries are needed.

Q5: Can this be used for probability calculations?
A: Yes, it can help determine the denominator in probability problems where order doesn't matter and items can repeat.