What is the binomial theorem?

The binomial theorem is a mathematical concept that provides a formula for expanding the powers of a binomial. A binomial is a mathematical expression with two terms, usually in the form of (a+b)n,(a + b)^n, where aa and bb are constants and nn is a positive integer. The binomial theorem expresses the result of raising a binomial to any positive integer exponent.

Binomial theorem formula

The following formula can be used to expand (a+b)n,(a + b)^n,which is how the binomial theorem is commonly represented in summation notation:

This formula states that each term in the expansion is given by the binomial coefficient(nk),\binom{n}{k},multiplied by an−ka^{n-k}and bkb^k, wherekkranges from 00 to nn.

Binomial coefficient

The binomial coefficient(nk)\binom{n}{k} is also known as n choose k or the number of techniques to choose kk elements from a set of nn elements without regard to the order. It is calculated using the formula given below:

Note: Where "n!n!" represents the factorial of nn.

Binomial theorem
Binomial theorem

Implementation

A simple implementation of the binomial theorem is as follows:

def calculate_binomial_coefficient(n, k):
    if k == 0 or k == n:
        return 1
    return calculate_binomial_coefficient(n - 1, k - 1) + calculate_binomial_coefficient(n - 1, k)
def expand_binomial(a, b, n):
    result = []
    for k in range(n + 1):
        coefficient = calculate_binomial_coefficient(n, k)
        term_a = a ** (n - k)
        term_b = b**k
        result.append(f"{coefficient} * {term_a} * {term_b}")
    return " + ".join(result)
# Driver code
a, b, n = 2, 3, 4
expanded_form = expand_binomial(a, b, n)
print(expanded_form)

Explanation

  • Lines 1–4: Uses recursion by the binomial_coefficient function to determine the binomial coefficient. It has two cases: a recursive case that executes the function with modified parameters and a base case that occurs when k is 0 or equal to n.

  • Lines 7–14: Three parameters, a, b, and n, are required by the expand_binomial function. It computes the binomial coefficient and the powers of a and b for the current k by iterating through each value of k from 0 to n (inclusive). For every term, it creates a string and adds it to the result list.

  • Lines 18–20: Sets the driver code values for a, b, and n, then calls the expand_binomial function with these values. The resulting expanded form is printed to the console.

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