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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

*:*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:

$$$$

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