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.
For integers we define
and call such numbers binomial coefficients.
- 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: