Counting Possible combinations

Question

I do not now how to count the number of possibilities. Perhaps you can help.

I have 12 sided dice. The sides are colored. 4 yellow 3 blue 2 green 1 red 1 black 1 white

If I roll 2 dice there are 16 possible combinations of two yellow sides showing out of a possible 144 combinations.

If I add dice, how do I count the number of possible yy combinations?

To complicate matters, the black dice are sometimes wild. When they are, then I want to count the number of yellow and black.

If I have two dice, then I want to count the possible combinations of yy, yb, by, and bb. I can make a simple table and observe the number of instances. In this case, there are 25 possible combinations out of 144 possibilities.

When I have more than 2 dice how many yy combinations are possible given x dice? And if I add the black dice how many yy, by, yb, bb combinations are possible given x dice?

Is there a formula that I can use to count the various combinations? Tables become impractical when there are more than 2 dice. I am unable to find formulas that help in counting the number of specific combinations our of the total number of combinations. In most examples, the identification of subset# out of total# seems to be done manually.

Answer 1

In the cases you've described, you just multiply the possibilities. Just as you got $16=4\cdot4$ YY combinations, with $3$ dice you would have $64=4\cdot4\cdot4$ YYY combinations. Similarly with four dice, there would be $4^4$ ways of getting all yellow.

For different colors of dice, you again multiply the possibilities. For blue and yellow, three are $3\cdot4$ possibilities, and for yellow and blue there are $4\cdot3$ possibilities. Usually, we would only do one multiplication and double the result.

When you have many dice, it would be very arduous to list all the possibilities. There's a short cut using binomial coefficients, which I gather you are unfamiliar with. There's a ton of stuff about it on the web.

Answer 2

This following is really a longish comment, not a solution, because it's not really feasible for large numbers of dice, but I think it's interesting to note that if we define $$P=4 y + 3 b + 2 g + r + b + w$$ then if we expand $P^2$ we have counted all the possible outcomes of rolling two dice: $$P^2 = 16 b^2+16 b g+8 b r+8 b w+32 b y+4 g^2+4 g r+4 g w+16 g y+r^2+2 r w+8 r y+w^2+8 w y+16 y^2$$ I.e. there are $16$ ways to roll blue-blue, $16$ ways to roll blue-green, $8$ ways to roll blue-red, etc. And in general, expanding $P^n$ counts all the outcomes when rolling $n$ dice. This process is only feasible for fairly small values of $n$, although it a computer algebra system can expand the range of application beyond the limits of pencil and paper computation.