## How to Use the Binomial Theorem

The binomial theorem is useful for working out variables to the n^{th} power. For example, to work out (a+b)^{3} by hand, you would need to use the distributive property on (a+b)^{.}(a+b)^{.}(a+b), which equals a^{3}+3a^{2}b+3ab^{2}+b^{3}. Figuring out (a+b)^{10} or (a+b)^{100} would take you all day to solve. That’s where the binomial theorem comes in handy. The theorem is stated as:

It looks complicated, but it saves a *lot* of time. Once you’ve worked an example or two, you’ll see how easy it is to actually use.

**Sample problem: **What is (x + y)^{4}? (Use the binomial theorem).

Step 1: **Insert the exponent into the binomial formula**, replacing “n”. The exponent in the question is 4, so:

The summation sign means that you are going to sum a series of parts — in this case the parts will be 4…3…2…1…0. If you had a power of 5, the parts would be 5…4…3…2…1…0 and so on.

Step 2: **Sum the series of parts**. The changes are highlighted in red and blue so you can see them more easily:

This looks complicated by all you are doing is:

- reducing the exponent for “x” by 1 in each step
- increasing the exponent for “y” by 1 in each step
- decreasing the number of items you choose in each step (indicated by the bottom number in parentheses).

Step 3: **Use combinatorics to expand the combinations part of the equation.** If you aren’t familiar with combinatorics, this combinations calculator walks you through the steps.

Step 4: **Evaluate the factorials.** This article on factorials will help if you aren’t familiar with factorials.

=x^{4} + 4x^{3}y + 6b^{2}y^{2} + 4xy^{3} + y^{4}

*That’s it!*

**Tip: **Note that 0! = 1, and x^{0} or y^{0}=0.

*Cheating*at Statistics? This is the Statistics Handbook that your professor

*doesn't want you to see*. So easy, it's

**Practically Cheating**. Find out more »