Applications of Pascal's Triangle

Probability and Combinations

Pascal's Triangle can be used for probability purposes. The book that Pascal wrote, Traite du triangle arithmetique (Treatise on the Arithmetic Triangle), used the triangle to solve probability problems (Jerphagnon). Combinations are frequently used in probability to find the total number of ways something can happen. These combinations are also called choose functions, and they are integral to finding binomial coefficients which are mentioned in the next section. Combinations uses the notation "\(\displaystyle {n \choose r}\)" to show the number of ways to select \(r\) objects from a collection of \(n\) objects. The terms in Pascal's Triangle align with these values. For example, the value for \(\displaystyle {n \choose r}\) is found in the r-th term in the n-th row of the triangle.

Let's look at an example of this in probability. Consider drawing two cards from a standard deck of cards. What is the probability that they are from the same suit? The probability can be found as follows: \[P(\text{same suit})= \frac{\text{Number of total ways to draw two cards from the same suit}}{\text{Number of total ways you can draw 2 cards from a standard deck}}. \]

The denominator can be found easily using a choose function. There are 52 cards and we are choosing 2 cards, so \(\displaystyle {52 \choose 2}\). To find this number using Pascal's Triangle, go to the 52nd row, and the 2nd term in the row. This number is 1326.

For our numerator, our desired result is to choose two cards from the same suit. There are 13 cards in each suit so the number of ways to choose two cards from a certain suit is \(\displaystyle {13 \choose 2}\). To find this number, go to the 13th row and the 2nd digit in the row. This number is 78. However, this only shows the number of ways to choose from a certain suit, but you could choose two cards of the same suit in any of the suits. So, there are \(\displaystyle {13 \choose 2}\) ways to choose two hearts + \(\displaystyle {13 \choose 2}\) to choose two spades + \(\displaystyle {13 \choose 2}\) ways to choose two clubs + \(\displaystyle {13 \choose 2}\) ways to choose two diamonds. Adding all of these together gives the total number of ways you can draw two cards from the same suit. The numerator then becomes \(4 \times \displaystyle {13 \choose 2}=4 \times 78=312.\)

Now, we can put these two things together to find our probability. \[P(\text{same suit})= \frac{312}{1326}= .235 \]

Binomial Theorem and Expansion

The Binomial Theorem is as follows:
For any non-negative integer n, the following is true: \[ (x+y)^n=\sum_{r=0}^n \displaystyle {n \choose k} \times x^{n-r} \times y^r. \] (Byju's)

The extension of the above theorem is called binomial expansion. The "\(\displaystyle {n \choose c}\)" portion of the expression is known as the "binomial coefficient." This notation is displaying the number of ways to select \(r\) objects from a collection of \(n\) objects. There is a formula to calculate these coefficients, but it is not necessary if Pascal's Triangle is used. Remember, each row in the triangle is named with the top row being row 0, and so on until row \(n\).

Consider a row \(n\) in Pascal's Triangle. The digits in this row are as follows: \(\displaystyle {n \choose 0}\), \(\displaystyle {n \choose 1}\), \(\displaystyle {n \choose 2}\), ... \(\displaystyle {n \choose n}\). This matches the binomial coefficients for the expansion of an expression of the form \((x+y)^n\).

To use Pascal's Triangle to expand binomials, use the Binomial Theorem to expand as shown above. For the coefficients, go to the \(n\)-th row in Pascal's Triangle. The first term in your row is your first coefficient, the second term is your second coefficient, and so on. The triangle is helpful, especially for students, for finding the coefficients in a simple way.

Multiple disciplines use the triangle to make new connections and find new patterns, and it is helpful for students who are learning probabilities and how to work with binomials.