Do you know the Fibonacci sequence, the Hockey stick Identity, and Pascal’s Identity? Well, all of these things are found in Pascal’s Triangle. What is Pascal’s Triangle you may ask? Pascal’s triangle is a triangle of numbers that goes on forever starting with 1, to 1 1, to 1 2 1, etc. But this triangle is no ordinary triangle, there are many cool patterns and formulas you can notice just from this triangle.
There are a lot of patterns in Pascal’s triangle, but there are a few that I was amazed by. The first one is the Fibonacci sequence. The Fibonacci sequence is formed by adding the 2 numbers before it, starting with 0, 1. So a part of the Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8. In Pascal’s triangle, if you go in diagonals, you will find out that it goes 1, 1, 2, 3, 5, 8, 13, etc. Another pattern is the powers of 11. If you look at each row, the first row is 11⁰, which is 1, the second is 11, 11¹, the 3rd row is 11², 121, the fourth is 1331 and the fifth is 14641. If you look at the 5th row, it is supposed to be 15101051 which is not 11⁵, but if you make it so the double-digit numbers overlap, it will make 161051, which is 11⁵. The same goes for line 6.
The last pattern I found impressive is the hockey stick. The hockey stick is how every diagonal is equal to the number bottom right to it starting from one of the ones on the side. One example is 1, 3, 6, 10, and 15 is equal to 35 as seen in the picture below. There is even a formula for this too which is
From Pascal’s triangle, you can create a formula that’s called Pascal’s identity.
If you look at Pascal’s triangle, you can see that every number, other than the ones on the sides, is the 2 numbers added above it. An example is 15, which is made up of 10 and 5, the 2 numbers above it. Now when you look at this you might think, “Isn’t this just a coincidence?” but there is a better way of thinking about this. Let’s say there are 10 people in total and you need to form a group of 3 chosen at random. This is c(10, 3). But then if you think of it as there are 2 different types of groups, one that includes Jack, one of the people in the group, and one that doesn’t. Well for the case that Jack is on it, there are c(9, 2) ways for choosing the other 2 people in the group, and for the case that Jack isn’t on it, there are c(9,3). That means that c(9, 2)+c(9,3) is equal to c(10,3). Does this sound familiar? Well, it’s Pascal’s identity just filled in with numbers. What this is telling us is that when you have n people and you have to choose k of them, it is equal to one person being in the group of k people plus one person definitely not being in the group of k people. And that’s Pascal’s identity.
Pascal’s triangle is an important and interesting thing and I am happy I learned about it. There are many patterns and formulas that come out of such a simple triangle and it helps me understand more about combinations. I have never seen this simple triangle have such complex formulas and patterns in it. This is why I love Pascal’s triangle.
