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.

1. is in integer
2. 
3. (Pascal’s rule).
4. for all n.
5. 
6. for n>0
7. 

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