Binomial theorem
You write the binomial theorem with combinations as
where n is a natural number.
Explanation
We calculate the different powers and see
⁰ = 1.gif){kind=small}
¹ = a + b.gif){kind=small}
² = a² + 2ab + b².gif){kind=small}
³ = a³ + 3a²b + 3ab² + b³.gif){kind=small}
⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.gif){kind=small}
⁵ = a⁵ + 5a⁴ b + 10a³ b² + 10a² b³ + 5a b⁴ + b⁵.gif){kind=small}
⁶ = a⁶ + 6a⁵ b + 15a⁴ b² + 20a³ b³ + 15a² b⁴ + 6a b⁵ + b⁶.gif){kind=small}
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