Now we are going to see about the binomial coefficient in Python. Here we will learn a lot of methods to calculate the binomial coefficients. In mathematics, binomial helps us to expand some terms with higher power easily. For example, if we have a number 103 to the power of 7. At that time, binomial is useful to expand this term. A binomial is known as a polynomial of the sum or difference of two terms.

**The binomial coefficient is a positive integer. It means is a positive whole number that is a constant in the binomial theorem.** **This is useful to expand the highest power. In mathematics, it is one of the most interesting and beneficial.**

## What is the formula for the binomial coefficient?

The formula for the binomial coefficient is

In this, the value of n should always be greater than ‘k’.

Let us see how to calculate the binomial coefficient in python in different functions.

**Here we are going to calculate the binomial coefficient in various functions they are:**

**scipy.special.comb()****scipy.special.binom()****math.combo() function****math.fact() function****using operator**

## Method 1: Finding Python Binomial Coefficient Using **scipy**.**special.comb()**

### What is the scipy module?

Scipy is a python library. It is useful for mathematical and scientific problems. Scipy is open-source. It is a built library in NumPy. scipy has some sub-packages. Now we are going to use the sub-packages to calculate the binomial coefficient. scipy.binom() and scipy.comb() are the sub-packages we will use.

First, we will see about scipy.comb() function to check the binomial coefficient.

### Syntax for scipy.comb()

scipy.special.comb(val1,val2)

### Parameter

val1- a value of n (must be greater than k) val2-value of k

### Returns

binomial coefficient

### Program

```
import scipy.special
print(scipy.special.comb(20,10))
```

First, we are importing a library as scipy. special. This module holds the attribute comb, next to giving 20 and 10 to get the binomial coefficient.

**Output**

184756.0

Next using scipy.binom() module to calculate the binomial coefficient.

## Method 2: Finding Python Binomial Coefficient Using **scipy.special.binom()**

### Syntax for scipy.binom()

scipy.special.binom(val1,val2)

### Parameter

val1- value of n (must be greater than val2) val2-value of k

### Returns

binomial coefficient

### Program

```
import scipy.special
print(scipy.special.binom(20,10))
```

First, we are importing a library as scipy.special. This module holds the attribute binom, next to giving 20 and 10 to get the binomial coefficient.

**Output**

184756.0

## Method 3: Finding Python Binomial Coefficient Using **math.combo() function**

### What is math.comb()?

The math module has a comb function that is used to calculate the binomial coefficient.

### Syntax

math.comb(val1,val2)

### Parameter

val1- a value of n (must be greater than k) val2-value of k

### Returns

binomial coefficient

### Program

```
import math
print(math.comb(20,10))
```

First, we are importing library math. Next, giving 20 and 10 to calculate the binomial coefficient.

**Output**

184756

## Method 4: Finding Python Binomial Coefficient Using **math.fact() function**

### What is math.fact()?

The math module has the fact() function to calculate the binomial coefficient.

### Program

```
from math import factorial as fact
def binomial(a,b):
return fact(a) // fact(b) // fact(a-b)
print(binomial(20,10))
```

First, we are importing the math functionâ€”next, declaring a function named binomial. Now giving parameters a and b. And then returning a formula to calculate the binomial coefficient.

**Output**

184756.0

**Recommended Reading |** Python Program for Factorial of a Number

## Method 5: Finding Python Binomial Coefficient Using Operator

```
import math
import operator
from functools import reduce
prod = lambda x,y:reduce(operator.mul, range(x, y+1), 1)
a= 20
b = 10
c=prod(b+1, a) / prod(1, a-b)
print(c)
```

First, importing math function and operator. From function tool importing reduce. A lambda function is created to get the product. Next, assigning a value to a and b. And then calculating the binomial coefficient of the given numbers.

**Output**

184756.0

## A fast way to calculate binomial coefficient in Python

```
def binomial(n, k):
if 0 <= k <= n:
a= 1
b=1
for t in range(1, min(k, n - k) + 1):
a *= n
b *= t
n -= 1
return a // b
else:
return 0
print(binomial(20,10))
```

First, create a function named binomial. The parameters are n and k. Giving if condition to check the range. Next, assign a value for a and b as 1. Now creating for loop to iterate. **floor division** method is used to divide a and b. Next, calculating the binomial coefficient.

**Output**

184756

## Finding Binomial Coefficient in Python Using Recursion

**Code**

```
def factorial(z):
if z==1:
return 1
else:
return z* factorial(z-1)
def binomial_coefficient(n,k):
a= (factorial(n)) / (factorial(k) * factorial(n-k))
return a
n=10
k=5
print("The binomial coefficient is:",binomial_coefficient(n,k))
```

The above code is calculating the binomial coefficient using recursion. First, we are creating a function named factorial. We all know that factorial is one of the best examples of recursion. And below, we are doing the calculation for factorial.

Next, create another function named binomial_coefficient on the next line using the formula to calculate the binomial coefficient. Giving the value of n and k. And at last, calculating the binomial coefficient.

But the above code is only useful for small numbers. If we want to go like the greatest numbers, we have to set the recursion limit.

**Output**

The binomial coefficient is:252.0

## Using recursion limit

As we already said, for the greatest numbers, we have to set the recursion limit. What we are going to do now:

**Code**

```
import sys
sys.setrecursionlimit(3000)
def factorial(z):
if z==1:
return 1
else:
return z* factorial(z-1)
def binomial_coefficient(n,k):
a= (factorial(n)) / (factorial(k) * factorial(n-k))
return a
n=1000
k=500
print("The binomial coefficient is:",binomial_coefficient(n,k))
```

First, we are going to Import a sys module. Sys module is to set the recursion limit. We are setting the recursion limit as 3000 so that we can calculate to 3000. Creating a function named factorial.

We all know that factorial is one of the best examples of recursion. And below, we are doing the calculation for factorial.

Next, create another function named binomial_coefficient on the next line using the formula to calculate the binomial coefficient. Giving the value of n and k. And at last, calculating the binomial coefficient.

**Output**

The binomial coefficient is:2.7028824094543655e+299

## Frequently Asked Questions Related to Binomial Coefficient Using Python

**1. What are the possible way to calculate the binomial coefficient?**

scipy.comb(), scipy.binom(),math.comb(), and math.fact() are the possible ways to calculate binomial coefficient.

**2. What is the usage of binomial coefficients?**

It is useful for analysis, and also it is a base for the binomial distributions.

**3. What is the way of setting the recursion limit?**

First importing **sys **module. And then setting the limit like **sys.setrecursionlimit()**.

## Conclusion

Here we have learned a lot about binomial coefficients. It is one of the interesting parts of mathematics. We can use binomial coefficients to expand the difficult term. It is advantageous and easy. The above-shown methods are the possible ways to calculate the binomial coefficients in Python.