Combinations of colored cubes Given three cubes and three colors (say red, green, and blue) I would like to calculate how many different ways I may associate colors with the cubes, not counting options like BGG and GBG twice. I have enumerated all the possibilities, I found it to be 10. I was wondering if there was a more general way to calculate it, if, I say, had x cubes and y colors.
 A: For small numbers like yours, any reasonably systematic method of listing and counting is a good way to handle the problem. So the answer below is useful mainly for larger numbers of cubes and/or colours. 
Think of the colours of paint as distinct bins, and of the cubes as identical balls. Choosing a colour scheme is equivalent to placing balls in bins. For example, the colour scheme BBG means placing $2$ balls in the Blue bin and $1$ in the Green bin. 
There is a standard method, often called Stars and Bars for solving such problems. If we have $x$ cubes, and $y$ colours of paint, that is, $x$ identical balls and $y$ distinct bins, then there are
$$\binom{x+y-1}{x},\quad\text{or equivalently}\quad \binom{x+y-1}{y-1}\tag{1}$$
to place the balls in the bins.
For your case $x=y=3$, this gives $\binom{5}{3}$, that is, $10$ ways.
We will not write out a justification of Formula 1, since the Wikipedia article linked to above is quite good. In addition, there are many many questions and answers on MSE that use Stars and Bars.
