Pascal's identity

Imagine you have a big jar of marbles with different colors. You want to know how many ways you can pick a certain number of marbles from the jar.

Pascal's identity is a fancy way of saying that if you add up the number of ways you can pick a certain number of marbles from the jar and the number of ways you can pick one fewer marble from the jar, you will get the number of ways you can pick the original number of marbles plus one extra marble from the jar.

It's like this:

Let's say you want to know how many ways you can pick 3 marbles from the jar. You can count them: one way to pick a red, a blue, and a green; another way to pick a green, a yellow, and a purple; and so on. Let's say you find out there are 10 ways to do this.

Now you want to know how many ways you can pick 4 marbles from the jar. To figure this out, you can add up the number of ways you can pick 3 marbles from the jar (which we know is 10) and the number of ways you can pick 2 marbles from the jar. Why 2 marbles? Because if you have already picked 3 marbles, you just need to pick one more to get to 4.

Let's say you find out there are also 10 ways to pick 2 marbles from the jar. Then, by Pascal's identity, you know there are 20 ways to pick 4 marbles from the jar.

This may seem like a bunch of math talk, but it's actually a really helpful idea that comes up in a lot of different situations. Whenever you want to know how many ways you can do something with different amounts or conditions, Pascal's identity can help you figure it out!