C# Coding Test Course, Finding Binomial Coefficients 2

Hello, everyone! In this lecture, we will delve deeper into binomial coefficients. The binomial coefficient is an important concept in combinatorics, representing the number of ways to choose k objects from n given objects. Particularly, it is a useful topic for preparing for coding interviews using C#.

Problem Description

Given natural numbers n and k, calculate the binomial coefficient C(n, k). The binomial coefficient is defined as follows:

C(n, k) = n! / (k! * (n-k)!)

Here, n! represents the factorial of n. That is, n! = n × (n – 1) × (n – 2) × … × 1. When calculating the binomial coefficient, the ranges of n and k are given as 0 to 30.

Input and Output Format

Input: The first line contains two integers n and k.

Output: Print the value of C(n, k).

Example

Input:
5 2

Output:
10

Algorithm Approach

To solve this problem, various methods can be used to calculate binomial coefficients. The most basic method is to use recursion. However, since recursion might be inefficient in terms of performance, we will look at using dynamic programming (DP) with memoization.

1. Dynamic Programming

We can construct a DP table to calculate the binomial coefficient, leveraging previously computed values to avoid redundant calculations. Specifically, we build the DP table using the following recurrence relation:

C(n, k) = C(n-1, k-1) + C(n-1, k)

The base cases are as follows:

  • C(n, 0) = 1 (There is exactly one way to choose 0 items from any n)
  • C(n, n) = 1 (There is also exactly one way to choose all n items)

2. C# Coding

Now, let’s write the code to calculate the binomial coefficient using dynamic programming in C#.

using System;

class Program
{
    static void Main(string[] args)
    {
        string[] inputs = Console.ReadLine().Split(' ');
        int n = int.Parse(inputs[0]);
        int k = int.Parse(inputs[1]);
        
        long result = BinomialCoefficient(n, k);
        Console.WriteLine(result);
    }

    static long BinomialCoefficient(int n, int k)
    {
        long[,] dp = new long[n + 1, k + 1];

        for (int i = 0; i <= n; i++)
        {
            for (int j = 0; j <= Math.Min(i, k); j++)
            {
                if (j == 0 || j == i)
                {
                    dp[i, j] = 1;
                }
                else
                {
                    dp[i, j] = dp[i - 1, j - 1] + dp[i - 1, j];
                }
            }
        }
        return dp[n, k];
    }
}

Code Explanation

The above C# code follows these steps:

  1. It takes input values for n and k from the user.
  2. It calls the BinomialCoefficient function to calculate the binomial coefficient.
  3. Within this function, it defines a 2D array dp to store the value of C(n, k).
  4. It systematically calculates all combinations and ultimately returns dp[n, k].

Time Complexity

The time complexity of this algorithm is O(n * k). This is because we need to iterate over all n and k to fill the DP table. The space complexity is also O(n * k), as space is required to store the DP array.

Conclusion

In this lecture, we learned how to calculate the binomial coefficient using C#. Learning how to solve problems efficiently using dynamic programming is also an important aspect. It is crucial to acquire various methods and thinking strategies when solving algorithm problems, so I encourage you to practice by tackling different problems and improving your skills.

Thank you!