Paul is really into playing with Lego, and he’s even more passionate about collecting Lego sets. He decided to collect each and every structure that it is possible to build with Lego. He built a set of drawers for his collection, and he is planning to store each possible construction in a separate compartment.
You need to convince him that it will take a great many compartments to do that, even if he limits himself to just a few Lego pieces.

And this is where my question comes in: How many ways are there of combining two black 2x4 Lego bricks? In other words, how many compartments does Paul need to fit all possible structures built out of two black 2x4 pieces?

Just to give you an idea, here are pictures of two possible structures built out of two Lego bricks.

Two constructions count as different when it is not possible to just turn one around and get the other. So the two structures below are actually the same:

Before you scroll down, grab a pair of Lego bricks and try to figure out how many different ways there are of putting them together.

So, have you figured it out?

Last chance to try it out for yourself…

Ok, there is a total of 24 possibilities for two pieces. Here they are:

And what is the number of possibilities for three pieces…?
Mind you, this one is really tough!
You can find the answer below.

These are the numbers that you get when you want to combine even more pieces:

1: 1
2: 24
3: 1560
4: 119580
5: 10166403
6: 915103765
7: 85747377755
8: 8274075616387

With just seven pieces there are ten times more possibilities than there are people on this planet.