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Binomial theorem

You write the binomial theorem with combinations as

where n is a natural number.

 


Explanation

We calculate the different powers and see









Let us try to find a logical structure for it. Every term can be written in the format

coefficient × am bn

The development of the coefficients is according the triangle of Pascal.

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1

A number is the sum of the number on the left and the right, just above it. In the rows these numbers increase at first, and decrease accordingly afterwards. That is the same as for calculating combinations with the formula

The upper number n is the power of the binomium, the lower number k is the current number of the term in the result. So we can write

The sum of the binomium is

Decomposition gives

A frequently used form shows the binomium as a series

 


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