1. What is Maximum Value of nCr When N is Even?

The maximum value of nCr (binomial coefficient) occurs when r = n/2 for even values of n. This represents the largest number of ways to choose a subset from a set of n distinct items.

2. How Does the Calculator Work?

The calculator uses the combination formula:

Where for maximum value when n is even:

• \( n \) — Total number of items (even integer)
• \( r = \frac{n}{2} \) — Number of items to choose
• \( ! \) — Factorial function

Explanation: The binomial coefficients are symmetric and reach their maximum at the center when n is even.

3. Importance of Maximum nCr Calculation

Details: Understanding maximum binomial coefficients is crucial in combinatorics, probability theory, statistics, and various optimization problems where you need to find the most probable or most numerous outcomes.

4. Using the Calculator

Tips: Enter an even positive integer for n. The calculator will compute C(n, n/2) which gives the maximum value of nCr for that particular n.

5. Frequently Asked Questions (FAQ)

Q1: Why does nCr reach maximum at n/2 when n is even?
A: The binomial coefficients are symmetric (C(n,r) = C(n,n-r)) and form a unimodal sequence that peaks at the center when n is even.

Q2: What if n is odd?
A: When n is odd, the maximum occurs at both floor(n/2) and ceil(n/2), and both values are equal.

Q3: What are some practical applications?
A: Used in probability calculations, statistical analysis, combinatorial optimization, and understanding distribution patterns in various scientific fields.

Q4: Are there limitations to large n values?
A: For very large n, factorial calculations can become computationally intensive and may require approximation methods like Stirling's formula.

Q5: How is this related to the binomial distribution?
A: The maximum nCr value corresponds to the most probable outcome in a binomial distribution with p = 0.5.