## Binomial Theorem

Can you expand on ? I guess you would find that is quite easy to do. You can easily find that .

How about the expansion of . It is no longer easy.

It is no longer easy isn’t it. However, if we use Binomial Theorem, this expansion becomes an easy problem.

Binomial Theorem is a very intriguing topic in mathematics and it has wide range of applications.

## Theorem

Let  be real numbers (or complex, or polynomial). For any positive integer , we have:

where,

Proof:

We will use prove by induction. The base case  is obvious. Now suppose that the theorem is true for the case , that is assume that:

we will need to  show that, this is true for



Let us consider the left hand side of equation above



## We can now apply Pascal’s identity:



The equation above can be simplified to:

as we desired.

### Example 1:  Power rule in Calculus

In calculus, we always use the power rule that

We can prove this rule using Binomial Theorem.

Proof:

Recall that derivative for any continuous function f(x) is defined as:



Let  be a positive integer and let

The derivative of f(x) is:

### Example 2:  Binomial Distribution

Let X be the number of Head a sequence of n independent coin tossing. X is usually model by binomial distribution in probability model. Let  be the probability that a head show up in a toss, and let . The probability that there is  head in the sequence of  toss is:



We know that sum of all the probability must equal to 1. In order to show this, we can use Binomial Theorem. We have:



Please also check another article Gaussian Samples and N-gram language models ,Bayesian model , Monte Carlo for statistics knowledges.