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 $(a + b)^n,$ where $a$ and $b$ are constants and $n$ 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 $(a + b)^n,$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$\binom{n}{k},$multiplied by $a^{n-k}$and $b^k$, where$k$ranges from $0$ to $n$.

Binomial coefficient

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

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

Implementation

A simple implementation of the binomial theorem is as follows:

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.