class="ltx_title_section" id="auto-label-section-403003">Pascal's Triangle

Suppose that class="ltx_title_section">Pascal's Triangle

While trying to solve a problem about gambling, Blaise Pascal discovered Pascal’s triangle (Benjamin, 2009 pg. 19). As we will see soon, this triangle relates to other topics in combinatorics quite heavily.

In digression, suppose that  you wanted to figure out the expansion of an equation such as contenteditable="false">\(\left(x+y\right)^2\).

We contenteditable="false">\(\left(x+y\right)^2\).We  may write this as follows contenteditable="false">\(\left(x+y\right)\left(x+y\right)=x^2+2xy+y^2\).

Now contenteditable="false">\(\left(x+y\right)\left(x+y\right)=x^2+2xy+y^2\).

Now  suppose we wanted to expand out \(\left(x+y\right)^3\).

We may write out \(\left(x+y\right)\left(x+y\right)\left(x+y\right)=x^2+xy+y^2\left(x+y\right)=x^3+3x^2y+3xy^2+y^3\).

Notice 

Notice that if we write the coefficients of these polynomials in the form of a triangle, we will see a pattern.

Suppose that you wanted to figure out the expansion of an equation such as \(\left(x+y\right)^2\).We may write this as follows \(\left(x+y\right)\left(x+y\right)=x^2+2xy+y^2\).

Now suppose we wanted to expand out \(\left(x+y\right)^3\).

We may write out \(\left(x+y\right)\left(x+y\right)\left(x+y\right)=x^2+xy+y^2\left(x+y\right)=x^3+3x^2y+3xy^2+y^3\).

Notice  that if we write the coefficients of these polynomials in the form of a triangle, we will see a pattern.

 

This is not a coincidence. The expansions of the form \(\left(x+y\right)^n\)  are actually predicted via the Binomial theorem (used to create   Pascal's triangle). To predict coefficients of these equations one can   use the formula for n choose k.

Example: Suppose you wanted to find the coefficient of \(x^6y^{10}\) of the expansion \(\left(5x+2y\right)^{50}\).

This is not a coincidence. The expansions of the form \(\left(x+y\right)^n\) are actually predicted via the Binomial theorem (used to create Pascal's triangle). To predict coefficients of these equations one can use the formula for n choose k.

Example: Suppose you wanted to find the coefficient of \(x^6y^{10}\) of the expansion \(\left(5x+2y\right)^{16}\).

You would simply write