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Binomial Theorem

By definition, the Binomial Theorem describes the algebraic expansion of powers of a binomial. The general binomial used in the formulation is typically (x  + y)^n, where x and y are variables and n is an integer greater than or equal to zero. However, to fully understand the use of the binomial theorem, we must first introduce the topic of Combinations. Now, as this chapter is not dedicated to counting problems, I'll just tell you in general what combinations are used for and then give you the formula.

A combination is a selection of items from a collection such that the order of selection does not matter. This is seen a lot in counting problems and sometimes in probability. Now, there are many notations for combinations, but the main one we'll see in this section is the first in the list below (although when typing formulas I will use the second notation).

This combination term is known in the Binomial Theorem as the binomial coefficient. This is the term that gives us the coefficient behind each term in the expansion of the binomial. The combination is defined as follows:

Where the "!" symbol is the factorial and k is less than or equal to n. Here's a list of some factorials just to get the basic idea of how they work.

0! = 1

1! = 1

2! = 2 * 1 = 2

3! = 3 * 2 * 1 = 6

4! = 4 * 3 * 2 * 1 = 24

5! = 5 * 4 * 3 * 2 * 1 = 120

In a more general sense,

n! = n * (n-1) * (n-2) * ...... * 3 * 2 * 1

Another way to see how the factorial works is to notice that it has a recursive definition.

n! = n * (n-1)! 

Where n is greater than or equal to 1 and 0! = 1.

Now that we have the binomial coefficient, we are ready to tackle the Binomial Theorem. I have written out the theorem in the image below.