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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 Theorem

Binomial Coefficient

For integers we define

and call such numbers binomial coefficients.

Properties.

  1. Binomial Coefficientis in integer
  2. (Pascal’s rule).
  3. for all n.
  4. 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

Binomial Theorem example

Binomial Theorem example 2

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

 

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