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Binomial expansion

In its simplest form the binomial theorem (bi means the 2 letters a and b) reads

 

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Explanation

We write the binomal expansion and see

Each term has the general form

coefficient × a^m bⁿ

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

							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. This you also see when calculating combinations. There the formula is

The upper number n is the power from the binomium, the bottom number k is the current number of the term. We can write

You can easily see that

what you can also write as

but that's just a matter of taste. In its simplest form the binomial theorem is

 

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