Welcome to the Interactive Twelvefold Way. This tutorial will guide you to use this app! (Press anywhere on the screen to continue)

The purpose of this app is to visualize how each formula has a certain number of configurations.

The goal is to find all the different ways you can put the balls into the bins.

When you put all the balls into the bins, press submit to record your submission.

Let's start with an easy example.

Go ahead and move any ball to any bin. Click and hold on a ball and drag it with your mouse to move it.

Great! Now go ahead and move the other ball to any bin.

Go ahead and press submit now!

Nice, you've gotten a correct configuration! (Press anywhere to continue)

Now, go ahead move ball #2 to the adjacent bin.

Go ahead and press submit.

Congrats! You scored another correct configuration! (Press anywhere to continue)

Notice how the ordering of the balls are different in the two configurations.

You want to find all of these different orders to find all the configurations of a formula!

Now go ahead and find the rest of the configurations. When you find all of them, the tutorial will resume.

Congratulations! You found all the configurations for this formula

For reference, this formula is $$n^k$$ If you plug n = 2 (# of bins) and k = 2 (# of balls), you will get 4. And now you understand how you get 4 at a more practical level.

Now let's try out a different formula. Click on the Formulas tab on the navbar to navigate to the formulas grid.

Now click on this formula: $${n+k-1 \choose n-1}$$

This formula uses the stars-and-bars model instead of the balls-and-bins model.

This formula uses the stars-and-bars model instead of the balls-and-bins model. The objective remains the same. You want to find all the different ways to put each bar into the gaps next to stars.

Think of it like this: You want to find all the different ways you can separate the stars into different groups.

Now let's find all the different ways. Go ahead and put the bar into any gaps (the gaps that have numbers below them).

Press submit when you place all bars into the gaps.

You got a correct configuration! Notice the different groups of stars. You have stars (or no stars) to the left of the bar and stars (or no stars) to the right.

Now go ahead and find the rest of the configurations. The tutorial will resume when you find them all!

Congratulations! You found all the configurations for this formula.

You are ready to solve the rest of formulas!

Certain formulas will have certain restraints and restrictions. It's up to you to figure it out!

For instance, shapes without numbered labels are identical, and thus their orders do not matter.

You are able change the number of balls and bins using the sliders to the left.

Refer to the general tips at the bottom for more information on terminologies!

That concludes the tutorial! Happy solving!