Ever wanted to know how to expand (a+b)¹⁸⁷? Well now you can!

### What is a Binomial Coefficient?

First, let’s start with a binomial. A binomial is a polynomial with two terms typically in the format (a+b)²

A binomial coefficient is raising a binomial to the power of n, like so (a+b)^n

We all remember from school that (a+b)² = a² + 2ab + b², but what is (a+b)⁸? This where the binomial formula comes in handy.

### Binominal Theorem

The Binomial Theorem is the expected method to use for finding binomial coefficients because it is how a computer would compute it. The theorem is as follows:

Luckily for us, this formula is the same as another formula we’ve seen, according to here.

The combinations formula! Let’s try an example.

#### Example

What is the coefficient of x⁶ in (1+x)⁸?

Simply plug this into the formula like so

Something that may confuse people is, how do we work out what n and k are? Well, we have n objects overall and we want to choose k of them. For binomial / combinatorics sums it helps to think “(combinations of) X taken in sets of Y” where x > y for obvious reasons, in this case “(combinations of) 8 taken in sets of 6”.

### Pascal’s Triangle

Pascal’s triangle is a triangle created by starting off with a 1, starting every line and ending every line with a 1 and adding the numbers above to make a new number; as seen in this gif.

No one could ever explain a maths topic as well as Numberphile, so here’s a Numberphile video on it:

#### Example

Let’s solve the example from earlier using Pascal’s triangle.