Pascal's triangle

Pascal’s triangle is filled with patterns that can solve many mathematical problems. One of the neat things that the triangle can do is help with binomial expressions. First let’s take a look at binomial expressions. Binomial expressions relate to the sum or difference of two terms such as:


Raising these expressions to powers of two can actually be pretty easy, for example:

\(\left(x+2\right)^2=x^2+2x+2x+4\) which then equals \(x^2+4x+4\). Yay we did it! Pretty easy. Now let’s try to take the binomial expression \(x+y\)  and raise it to the 4th power. Do we really want to do that by hand? Of course it can be done by hand but it is very tedious work. This is where Pascal’s triangle comes in handy. 

First we want to look at Pascal’s triangle and see how it is generated. To generate the triangle, we will write down the number one and then we move on to a new row where we write the number \(1\) twice. After this, we notice that each row must begin and end with the number one. To calculate the numbers in each row we will have to add the two numbers above that are on the left and right. You can create as many rows as you want.

We now can see how nice it is to use Pascal’s triangle when expanding binomial expressions. Each row of the triangle matches the coefficients of a binomial expansion of the form \(\left(x+y\right)^n\) where \(n\) is the number of the row. We start counting at the top of Pascal's triangle where the first number (number \(1\)) Is the row zero and the row containing two ones, is row \(1\).

Let’s try it for \(\left(x+y\right)^4\), by looking at row 4 of Pascal’s triangle we see that we obtain \(x^4+4x^3y+6x^2y^2+4xy^3+y^4\). We see that the coefficients match the row \(4\) of Pascal’s triangle.