Combinations, how many combos can you make out of these amount of things?

Question

I don't know why this is giving me trouble.

A generator is a cup that has 6 cubes, 3 are red and 3 are black. The red ones count for 0 points and the black count for 1 point.

You draw from each cup one time and then replace the cubes.

So if you have 0 generators you can only get a sum of 0. If you have 1 generator you can get a sum from 0 to 1. For 2 generators you can get a sum from 0 to 2....and so on.

How many combinations can you have for 1 to 4 generators?


This is what I have but I want to make sure I have the correct amount.

1 generator: B, R
2 generators: BB, BR, RB, RR
3 generators: BBB, BBR, BRB, RBB, RRR, RRB, RBR, BRR
4 generators: I don't want to write them out, but I have 16 all together

Is that right???? Thank you so much!

Answer 1

No.

First off, an easier way of doing this, without actually writing everything out, would be using math!!!

You have 2 choices (B or R) so to find out the number of combinations, you use:

2^(number of units in the combination, what you referred to as "generators")

So for the first one... its one generator... so 2^1 = 2 and that coincides with your answer...

for the second one, 2^2 = 4
we're still good

the third one, 2^3 = 8
still good!

now things change!
normally you could just continue this and you'd see that 2^4 = 16... however, look at what you wrote up above:

"3 are red and 3 are black"

there is a limit to how many of each object can exist! this was not an issue when using combinations of three or less, but now that we have reached 4, it is possible to use up all of one color! This makes the combination BBBB and RRRR both impossible. Therefore, for the fourth generator there should only be 14 (2^4 = 16 :: 16 - 2 = 14).

Answer 2

this is statistics stuff. you have 2 possible outcomes, B & R. If you take them one at a time, 2 to the power of 1, you get 2. If you take them 2 at a time, 2 to the power of 2 or 2*2 = 4, If 3 at a time, 2 to the power of 3 or 2*2*2=8. For 4 at a time, 2 to the power of 4 or 2*2*2*2=16

Answer 3

Yes