python coding test course, finding binomial coefficients 2

In this blog post, we will take a closer look at how to calculate the binomial coefficient using Python.
The binomial coefficient is an important concept in combinatorics that represents the number of ways to select r elements from a given n elements.
The binomial coefficient is defined by the following formula.

Definition of Binomial Coefficient

The binomial coefficient C(n, r) is defined as follows: 

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

Here, n! denotes the factorial of n, which is the product of all natural numbers from n to 1.
For example, 5! is 5 * 4 * 3 * 2 * 1 = 120. The binomial coefficient is very useful for calculating the number of combinations.

Problem Description

Write a function that calculates the binomial coefficient C(n, r) for given integers n and r.
n is a non-negative integer, and r is a non-negative integer less than or equal to n.

Input Format

The first line contains two integers n and r, separated by a space.

Output Format

Print the binomial coefficient C(n, r).

Example Input/Output

Input:
5 2
Output:
10

Solution Approach

There are several approaches to solve this problem. The most intuitive method is to directly calculate factorials using the mathematical definition.
However, calculating factorials directly can be inefficient for large n.
Therefore, we will use a dynamic programming (DP) approach to solve this problem more efficiently.

Calculating Binomial Coefficient Using Dynamic Programming

One of the properties of binomial coefficients is as follows: 

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

Using the above formula, we can decompose C(n, r) into previous binomial coefficients.
This can be implemented in the form of a DP table as shown below.

How to Create a DP Table

Define dp[i][j] as C(i, j). The following rules can be applied to compute all binomial coefficients:

  • dp[i][0] = 1 (for all i)
  • dp[i][i] = 1 (for all i)
  • Else dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j] (0 < j < i)

Python Implementation

Below is the Python code for calculating the binomial coefficient: 

def binomial_coefficient(n, r):
    # Initialize 2D DP table
    dp = [[0] * (r + 1) for _ in range(n + 1)]

    # Fill the DP table
    for i in range(n + 1):
        for j in range(min(i, r) + 1):
            if j == 0 or j == i:
                dp[i][j] = 1
            else:
                dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j]

    return dp[n][r]

# Example call
n, r = map(int, input().split())
print(binomial_coefficient(n, r))

Time Complexity Analysis

The above algorithm has a time complexity of O(n * r).
Since there are two nested for loops, in the worst case, it will handle all values of n and r.
However, it can operate very quickly for small ranges of n and r.
This algorithm is a good method to effectively calculate binomial coefficients.

Conclusion

In this post, we explored various methods to compute the binomial coefficient C(n, r).
From traditional factorial-based methods to dynamic programming approaches,
we examined the advantages and disadvantages of each approach.
Since binomial coefficient problems frequently appear in coding tests,
it is important to have a good understanding of this content.