# Binomial Theorem

## What is Binomial Theorem?

The expression of the from (a + b)” is called a binomial. By direct multiplication it is easy to expand (a + b)^{2} , (a + b)^{3 , }etc. The ancient mathematicians about the expansion of (a + b )^{n} for 0 _{n} 7. around 1660, B Pascal introduced Pascal’ s triangle for the coefficients in the expansion of (a + b )^{n} and in the same year he gave the present from of binomial theorem.

## Binomial Coefficient

For integers we define

and call such numbers binomial coefficients.

### Properties.

- is in integer
- (Pascal’s rule).
- for all n.
- for n>0

Properties 5 and 6 are the binomial theorem applied to and , respectively, although they also have purely combinatorial meaning.

## Binomial Theorem Examples

### Example 3: When the exponent, *n*, is 3.

The terms are:

k=0: | k=1: | k=2: | k=3: |
---|---|---|---|

a^{n-k}b^{k}= a ^{3-0}b^{0}= a^{3} |
a^{n-k}b^{k}= a ^{3-1}b^{1}= a^{2}b |
a^{n-k}b^{k}= a ^{3-2}b^{2}= ab^{2} |
a^{n-k}b^{k}= a ^{3-3}b^{3}= b^{3} |