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:
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.