Binomial Coefficient Algorithm
The Binomial Coefficient Algorithm is an efficient technique to compute the binomial coefficients, which are the coefficients of the terms in the expansion of a binomial expression. These coefficients can be represented using the symbol C(n, k) or as "n choose k", where n is a non-negative integer and k is an integer ranging from 0 to n. The binomial coefficients have a wide range of applications in mathematics, statistics, and computer science, including combinatorics, probability theory, and coding theory. They are essential in calculating the number of ways to choose k elements from a set of n elements without regard to the order.
The algorithm utilizes the properties of Pascal's Triangle, where each number in the triangle is the sum of the two numbers directly above it. The triangle starts with a single 1 at the top, and the subsequent rows are constructed by adding the numbers diagonally adjacent from the row above. The Binomial Coefficient Algorithm exploits the recurrence relation C(n, k) = C(n-1, k-1) + C(n-1, k) to compute the coefficients in a dynamic programming approach. By storing and reusing the previously computed values, the algorithm significantly reduces the time complexity compared to the direct calculation using the factorial formula. This approach can be further optimized with the use of memoization or iterative methods, ensuring a fast and efficient computation of binomial coefficients for a wide range of problems.
using System;
namespace Algorithms.Numeric
{
/// <summary>
/// The binomial coefficients are the positive integers
/// that occur as coefficients in the binomial theorem.
/// </summary>
public static class BinomialCoefficient
{
/// <summary>
/// Calculates Binomial coefficients for given input.
/// </summary>
/// <param name="num">First number.</param>
/// <param name="k">Second number.</param>
/// <returns>Binimial Coefficients.</returns>
public static long Calculate(int num, int k)
{
if (num < k || k < 0)
{
throw new ArgumentException("n ≥ k ≥ 0");
}
return Factorial.Calculate(num) / (Factorial.Calculate(k) * Factorial.Calculate(num - k));
}
}
}