1. What is the Binomial Coefficient?

The binomial coefficient C(n,k), read as "n choose k", counts the number of ways to choose k elements from a set of n elements without regard to order. It's a fundamental concept in combinatorics and appears in the binomial theorem.

2. How Does the Calculator Work?

The calculator uses the binomial coefficient formula:

Where:

• \( n! \) — Factorial of n (n × (n-1) × ... × 1)
• \( k! \) — Factorial of k
• \( (n-k)! \) — Factorial of (n-k)

Explanation: The formula calculates the number of distinct combinations by dividing the total permutations by the permutations of the selected items and the remaining items.

3. Applications of Binomial Coefficients

Details: Binomial coefficients are used in probability calculations, polynomial expansions (binomial theorem), statistical analysis, and combinatorial mathematics. They appear in Pascal's Triangle and have applications in computer science algorithms.

4. Using the Calculator

Tips: Enter non-negative integers where n ≥ k ≥ 0. For large values (n > 20), consider using a specialized mathematical software as factorials grow very rapidly.

5. Frequently Asked Questions (FAQ)

Q1: What if k > n?
A: By definition, C(n,k) = 0 when k > n, as you can't choose more items than you have.

Q2: What are some properties of binomial coefficients?
A: They are symmetric (C(n,k) = C(n,n-k)), and satisfy Pascal's identity: C(n,k) = C(n-1,k-1) + C(n-1,k).

Q3: How does this relate to probability?
A: In probability, C(n,k) gives the number of possible outcomes for "k successes in n trials" in combinations.

Q4: What's the largest n this calculator can handle?
A: Due to PHP's integer limits, n > 20 may give inaccurate results. For n=20, C(20,10) = 184756.

Q5: Are there alternative ways to compute binomial coefficients?
A: Yes, using Pascal's Triangle or multiplicative formulas can be more efficient for large numbers.