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
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: |
|---|---|---|---|
| an-kbk = a3-0b0 = a3 |
an-kbk = a3-1b1 = a2b |
an-kbk = a3-2b2 = ab2 |
an-kbk = a3-3b3 = b3 |