Binomial expansion

Binomial expansion uses an expression to make a series. It uses a bracket expression like formula_1. There are three binomial expansions.
The formulas.
There are basically three binomial expansion formulas:
We can explain why there are such 3 formulas with a simple expansion of the product:
Using Pascal's triangle.
If formula_5 is an integer (formula_6), we use Pascal's triangle.
To expand formula_7:
So formula_14
For example:
So as a rule:
where formula_17 is the number at row formula_5 and position formula_19 in Pascal's triangle.