1. What is the Combination Formula?

The combination formula calculates the number of ways to choose a subset of items from a larger set where order doesn't matter. When M specific items must always be included, the formula becomes: C((n-m), (r-m)).

2. How Does the Calculator Work?

The calculator uses the combination formula:

Where:

• \( n \) — Total number of items
• \( m \) — Number of specific items that must always be included
• \( r \) — Number of items to be selected
• \( C \) — Combination function (binomial coefficient)

Explanation: Since M specific items must always be included, we subtract them from both the total items and the selection count before calculating combinations.

3. Importance of Combination Calculation

Details: Combination calculations are fundamental in probability theory, statistics, and combinatorial mathematics. They help determine possible outcomes in various scenarios where order doesn't matter.

4. Using the Calculator

Tips: Enter positive integer values for N, M, and R. Ensure M < N, R < N, and M ≤ R. The calculator will compute the number of combinations where M specific items are always included.

5. Frequently Asked Questions (FAQ)

Q1: What does "M specific things always occur" mean?
A: It means that in every combination we count, all M specified items must be included in the selection of R items.

Q2: Why subtract M from both N and R?
A: Since M items are fixed in the selection, we only need to choose the remaining (R-M) items from the remaining (N-M) items.

Q3: What if M = 0?
A: If M = 0 (no specific items required), the formula simplifies to the standard combination formula C(n, r).

Q4: What if R = M?
A: If R = M, we must select exactly the M specific items, so there's only 1 possible combination.

Q5: Can M be greater than R?
A: No, M cannot be greater than R because we cannot include more specific items than we're selecting.