Hattori–Stong theorem
Okay kiddo, have you ever played with blocks before? Let's imagine that we have a lot of different colored blocks and we want to put them in a line. We can put them in any order we want, and each block can only be used once. This is called a "permutation" of the blocks.
Now, let's say we have x red blocks, y yellow blocks, and z green blocks. How many different permutations can we make with these blocks? Well, one way to figure it out would be to list all the possible permutations and count them up. But that could take a really long time if we have a lot of blocks!
Luckily, the Hattori-Stong Theorem helps us figure out how many permutations there are without having to list them all out. It works by using some fancy math that involves something called "factorials" (which is just a way of multiplying a sequence of numbers).
So the theorem tells us that the number of permutations we can make with x red blocks, y yellow blocks, and z green blocks is equal to:
x! * y! * z! / (x + y + z)!
Now, that might look like a bunch of gobbledegook to you, but it's actually pretty simple. The exclamation points just mean "factorial" and the slashes mean "divided by". So we're just taking the factorial of each of the numbers (which just means multiplying them together in a certain way) and then dividing by the factorial of the total number of blocks.
If that still seems a bit confusing, don't worry – the important thing to understand is that the Hattori-Stong Theorem helps us figure out how many permutations we can make without having to list them all out by hand. And that's a pretty neat trick!
Now, let's say we have x red blocks, y yellow blocks, and z green blocks. How many different permutations can we make with these blocks? Well, one way to figure it out would be to list all the possible permutations and count them up. But that could take a really long time if we have a lot of blocks!
Luckily, the Hattori-Stong Theorem helps us figure out how many permutations there are without having to list them all out. It works by using some fancy math that involves something called "factorials" (which is just a way of multiplying a sequence of numbers).
So the theorem tells us that the number of permutations we can make with x red blocks, y yellow blocks, and z green blocks is equal to:
x! * y! * z! / (x + y + z)!
Now, that might look like a bunch of gobbledegook to you, but it's actually pretty simple. The exclamation points just mean "factorial" and the slashes mean "divided by". So we're just taking the factorial of each of the numbers (which just means multiplying them together in a certain way) and then dividing by the factorial of the total number of blocks.
If that still seems a bit confusing, don't worry – the important thing to understand is that the Hattori-Stong Theorem helps us figure out how many permutations we can make without having to list them all out by hand. And that's a pretty neat trick!