Python Coding Test Course, Finding Binomial Coefficient 1

1. What is a Binomial Coefficient?

A binomial coefficient is defined in combinatorics for two integers n and k. It represents the number of ways to choose k items from n items, and is denoted as C(n, k) or (n choose k). The binomial coefficient is calculated as follows:

• C(n, k) = n! / (k! * (n-k)!)

Here, n! is the factorial of n, where n! = n × (n-1) × (n-2) × … × 1.
Binomial coefficients are very useful in solving combinatorial problems.

2. Problem Description

Problem: Write a function to calculate the binomial coefficient C(n, k) for given n and k.
n is an integer from 0 to 30, and k is an integer from 0 to n.

3. Problem Solving Approach

There are several methods to calculate the binomial coefficient.
We can use recursion, dynamic programming, or mathematical formulas to solve it.
Here, we will solve the problem using the Dynamic Programming approach.

3.1 Dynamic Programming

Dynamic programming is a method that divides the problem into smaller subproblems and saves the results of these subproblems for reuse in subsequent calculations. The binomial coefficient can be calculated using the following recurrence relation:

• C(n, k) = C(n-1, k) + C(n-1, k-1)
• Base cases: C(n, 0) = C(n, n) = 1

In the above recurrence relation, we can calculate the binomial coefficient recursively for each case, but this leads to redundant calculations. To avoid this, we will use a DP table for our lecture.

4. Implementation

Now, let’s write the code for a Python function to calculate the binomial coefficient.
We will implement the method using a DP table.

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

    # Set base cases
    for i in range(n + 1):
        dp[i][0] = 1  # C(i, 0) = 1
        dp[i][i] = 1  # C(i, i) = 1

    # Fill the DP table according to the recurrence relation
    for i in range(1, n + 1):
        for j in range(1, min(i, k) + 1):
            dp[i][j] = dp[i - 1][j] + dp[i - 1][j - 1]

    return dp[n][k]

            

In the above code, we initialize the DP table for the given n and k, set the base cases, and then fill the DP table using the recurrence relation.
As a result, the binomial coefficient is stored in dp[n][k].

5. Testing

It’s time to test the function we implemented above. We will use a simple example like C(5, 2) to verify the accuracy of the function.

# Test
print(binomial_coefficient(5, 2))  # Output: 10
print(binomial_coefficient(30, 15))  # Output: 155117520

            

By calling the function this way, we can calculate the binomial coefficient and print the results.
Check if the results for the above input values are correct.

6. Conclusion

In this lecture, we learned about the definition and calculation methods of binomial coefficients.
We learned how to efficiently calculate binomial coefficients using dynamic programming.
We’ve also looked closely at the implementation process using Python.
This algorithm is frequently used in actual coding tests and algorithm problem solving, so
make sure to master it.

Additionally, try solving advanced problems related to binomial coefficients to further improve your skills.
Thank you.