# Binomial Theorem Python – Printing the Binomial Series

In this tutorial, we will see how to implement the Binomial Theorem in Python and print the corresponding series for a given set of inputs.

We use **Binomial Theorem **in the expansion of the equation similar to (a+b)^{n}. To expand the given equation, we use the formula given below:

In the formula above,

- n = power of the equation
- a, b = terms with coefficients
- r = takes on the successive values from 0 to n
- C = combination and its formula is given as:

where,

- n = Number of Elements
- r = Number of Elements to be selected
- n >= r

For example:

(a+b)^{4} = ^{4}C_{0}a^{4}b^{0} + ^{4}C_{1}a^{4-1}b^{1} + ^{4}C_{2}a^{4-2}b^{2} + ^{4}C_{3}a^{4-3}b^{3 }+ ^{4}C_{4}a^{4-4}b^{4}^{ }

(a+b)^{4 }= a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}

## Algorithm for Binomial Theorem Python

- Declare a Function.
- Inside the function, take the coefficient of a and b and the power of the equation, n, as parameters.
- Calculate the first term by raising the coefficient of a to the power n. Subsequently, append it to the series list.
- Calculate the next term inside a for loop using the previous term. Use the formula mentioned below.

- Format and Append the next term to the series list.
- Convert the list to a string and print it.

We can implement an extra inner function for the formatting of the terms which would contain many if-else clauses. These clauses will decide how the terms will be formatted.

## Python Code: Print the Binomial Series

def form_series(co_a, co_b, n): """ This method creates the Binomial Theorem Series. :param co_a: coefficient of a :param co_b: coefficient of b :param n: power of the equation :return: None """ def formatting(next_term, coeffs): """ This is an inner function which formats the terms of the binomial series. :param next_term: coefficient of next term :param coeffs: powers of a and b :return: formatted term """ if next_term == 1: coeffs.insert(0, "") else: coeffs.insert(0, next_term) if coeffs[1] == "^0" and coeffs[2] == "^0": return coeffs[0] elif coeffs[1] == "^0": return "{}b{}".format(coeffs[0], coeffs[2]) elif coeffs[2] == "^0": return "{}a{}".format(coeffs[0], coeffs[1]) elif coeffs[1] == "^1" and coeffs[2] == "^1": return "{}ab".format(coeffs[0]) elif coeffs[1] == "^1": return "{}ab{}".format(coeffs[0], coeffs[2]) elif coeffs[2] == "^1": return "a{}b".format(coeffs[0], coeffs[1]) return "{}a{}b{}".format(coeffs[0], coeffs[1], coeffs[2]) # Initializing a list named as `series` series = list() # Calculating the First Term, Formatting it # and Appending it to our Series first_term = pow(co_a, n) coeffs = ["^" + str(n), "^0"] series.append(formatting(first_term, coeffs) + " + ") next_term = first_term # Calculating, Formatting and Appending # the remaining terms. for i in range(1, n + 1): # We can find next term using the # previous term and the formula # mentioned below. next_term = int(next_term * co_b * (n - i + 1) / (i * co_a)) # Pre-formatted list creation coeffs = ["" if x == 1 else "^" + str(x) for x in [n - i, i]] # Append till last term is not reached if i != n: series.append(formatting(next_term, coeffs) + " + ") # Append the last term. else: series.append(formatting(next_term, coeffs)) # Joining the series as a string and printing it. print("".join(series)) if __name__ == "__main__": # Taking inputs print("( a + b ) ^ n") co_a = int(input("Enter the coefficient of a: ")) co_b = int(input("Enter the coefficient of b: ")) n = int(input("Enter n: ")) print("({}a+{}b)^{} = ".format(co_a, co_b, n),end=" ") # Calling the Function form_series(co_a, co_b, n)

#### Input:

Binomial Theorem Python

#### Output:

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