Puzzle

We have a collection of red and blue beads. They can be arranged in many different patterns. Let's start with pairs of beads. How many different arrangements of two beads are possible if each bead may be red or blue?  

There are four possible arrangements. To create all four of the arrangements requires eight beads.

But if we put the beads on a string, it is possible to include all four arrangements within a string that uses fewer than eight beads. This string of five beads includes all four arrangements shown above.

Here's how:

How many different arrangements of three beads are there if, again, each bead may be red or blue?

Here are three possible arrangements.

There are more. How many arrangements of three red and blue beads are there, and how many beads do you need to make all of them? And, as we did with arrangements of two beads, can you make a single string of beads that includes all possible arrangements of three beads? How long is that string?

Try filling in this table:
 

Fill in the table as far as you want to go for larger and larger numbers of beads.

The last line is for a general solution. If you have n beads, each of which may be red or blue, how many possible arrangements are there? How many beads are needed to make all of those arrangements? How many beads are needed for a string that includes all of those arrangements?

Extra

What if you made a loop instead of a string in each case? How many beads would be required to include all the possible arrangements of each number of red and blue beads?