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 a wide range of applications.
Let , be real numbers (or complex, or polynomial). For any positive integer , we have:
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 the 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 the Binomial Theorem.
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 the probability model. Let be the probability that a head shows up in a toss, and let . The probability that there is head in the sequence of toss is: