Pascal’s Triangle was named after Blaise Pascal. Pascal’s triangle starts with the number 1 and goes down the scale. When you start with one, add more numbers in a triangular shape, like a pyramid of some sort. All the numbers on the surrounding right and left sides of the triangle are one. The insides of the triangle are then filled out by finding the sum of the two numbers above it to its left and right (Hosch, 2009, Pierce, 2014). The formula for Pascal’s Triangle is usually written in a form “n choose k” which looks like this: (Pierce, 2014). Pascal’s Triangle is also a never ending triangle of equilaterals (Coolman, 2015). The triangle is symmetric to the other side, with means if you divide the triangle in half, the numbers on the left are the exact same numbers on the right (Pierce, 2014). To find the numbers inside of Pascal’s Triangle, you can use the following formula: nCr = n-1Cr-1 + n-1Cr. Another formula that can be used for Pascal’s Triangle is the binomial formula.

What is the Binomial Theorem?

The binomial theorem is used to find coefficients of each row by using the formula (a+b)n. Binomial means adding two together. According to Rod Pierce, binomial theorem is “what happens when you multiply a binomial by itself… many times.” (2014.) Another way of finding a solution is using binomial distribution, which is like playing a game of heads and tails. The formula for binomial distribution is: .

The binomial formula is (a+b)n. The more complex version would be:

As you can see, the binomial formula equals the “n chooses k” formula (Pierce, 2014). Binomial Distribution has to do with Pascal’s Triangle in the sense that when the nth row (from (a + b)n) is divided by 2n, that nth row becomes the binomial distribution.

Coin Tosses in Relation to Binomial Theorem

When tossing a coin, there are two possible results, head or tails. There is a ½ chance of getting heads and a ½ chance of getting tails. In the event that we flip two coins, there are four (three) conceivable results. We may get two heads, or two tails, or one head and one tail (x2). The possibility of getting two heads is one out of four, or ¼. The shot of getting two tails is ¼. The shot of getting one head and one tail is two out of four, or ½ (Spencer, 1989). As shown in the table below, the toss would represent the row in Pascal’s Triangle.

The heads and tails method for row one is like flipping two coins and getting two results. The first row is organized, 1, for getting a tails, another 1 for getting heads, and 2 for the number of coins, as explained earlier gets the order of the first row, 1 2 1.

~ Heads/Tails Chart/Diagram

Other Patterns in Pascal’s Triangle

The coin toss might be one pattern, but there are others. Some others are the “horizontal sums” (Pierce, 2014). The horizontal sums pattern is adding up the numbers in each row and getting their sums. If you keep doing this, you see the pattern where the sum doubles at each row (Pierce, 2014).

Another pattern is the “exponents of 11” pattern (Pierce, 2014). In this pattern, first, you raise 11 to 0 (110), then you raise it to the numbers after 0 (for example, 110, 111, 112, 113…). The way this relates to Pascal’s Triangle is that 110 = 1, and the number in the first row in Pascal’s Triangle is 1. 111 = 11, and the numbers in the second row are 1, 1. 112 = 121, and the numbers in the third row are 1, 2, 1. This goes on so on and