Blog Post 2

\(1\) \(\underline{\text{Pascal's Triangle}}\)

In discrete mathematics, Pascal’s triangle is the arrangement of binomial coefficients in such a way that a triangle is formed. Although such a pattern was studied centuries before his time, we refer to Pascal’s triangle in relation to Blaise Pascal, a French mathematician. The triangle was originally developed by the ancient Chinese, but Pascal was the first person to discover the importance of all of the patterns that occur within it. His work allegedly stemmed from the popularity of gambling. After considering a question asked to him about gambling with dice, Pascal’s \(\textit{Arithmetical Triangle}\) resulted.

The triangle is created by starting at the top, row \(0\), with the number \(1\). Each row afterwards begins and ends with \(1\) and the pattern follows that as you move through the row, you add the number above and to the left with the number above and to the right for any given position. A portion of Pascal’s triangle is shown below.

Shown (above) is a part of Pascal’s triangle.

Notice that this illustrates how one can use the previous row to find the next values in the figure. The question then arises of how we would compute the values that lie in very large positions in the triangle. Fortunately, there is a formula from combinations for working out the value at any place in Pascal’s triangle. This formula is the familiar “\(n\) choose \(k\),” where \(n\) is the number of the row and \(k\) is the position in that row.

Pictured (above) is the formula for “\(n\) choose \(k\)” as well as how it can be applied to Pascal’s triangle.

There are numerous identities, patterns, and uses for Pascal’s triangle. One example that we will look at here is how it is used in algebra to find the coefficients in binomial expansions. As an illustration, consider the expansion \((x+y)^3=x^3+3x^2y+3xy^2+y^3\). Notice that the coefficients are \(1, 3, 3, 1\) and that these are found in the \(3\)rd row of Pascal’s triangle.

Additional patterns that are found within Pascal’s triangle include:

  • Fibonnacci’s Sequence

  • Triangular Numbers

  • Square Numbers