Polynomial method in combinatorics

The polynomial method in combinatorics is like counting stickers on a sticker sheet, but instead of stickers, we count numbers. Let's say you have a string of numbers, like 3, 1, 4, 1, 5, 9, and you want to count how many times the number 1 appears in the string.

To do this, we can create a special kind of math equation called a polynomial. It's like a bunch of stickers stuck together to make a bigger sticker. For instance, we can take the string of numbers and make a polynomial like this:

(3 + x)(1 + x)(4 + x)(1 + x)(5 + x)(9 + x)

The "+" in the equation means that we're adding stickers together, and the "x" is just a placeholder for a number we don't know yet.

Now let's say we want to count how many times the number 1 appears in the string of numbers. We simply plug in the number 1 for the "x" in the polynomial, like this:

(3 + 1)(1 + 1)(4 + 1)(1 + 1)(5 + 1)(9 + 1)

This simplifies to:

4 * 2 * 5 * 2 * 6 * 10

And if we multiply these numbers together, we get:

2400

So there are 2400 ways to choose the number 1 in the string of numbers.

We can use this idea to solve all sorts of combinatorics problems, like counting how many ways there are to color a map or arrange a set of objects. Instead of counting the actual objects, we create a polynomial that represents them, and then we plug in certain values to count the specific things we're interested in. It's like counting stickers, but with math!