You roll 3 dices with 6 sides each - Three "0", Two "1" and One "2" - what are the percentages of each (0-6) result?

Question

I'm making a game based on Dice Rolls, and I need some percentages based on the results of these rolls.

The situation: You roll 3 dice and get points based on the results. Each die has 6 sides - Three "0", Two "1" and One "2".

The Problem: Since you can get results that vary from 0-6, I need the chances of getting each one.

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What I did and what I need:

Result 0: (0,0,0) - 1/8 or 12,5%

Result 1:

Result 2:

Result 3:

Result 4:

Result 5:

Result 6: (2,2,2) - 1/216 or 0,46%

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Thanks!

Answer 1

This really just shows the answer, not really a "why." Wolfram alpha says:

$$(3+2x+x^2)^3=x^6+6 x^5+21 x^4+44 x^3+63 x^2+54 x+27$$

So your probabilities are: $$\begin{align} P(0)&=\frac{27}{216}=\frac{1}8\\ P(1)&=\frac{54}{216}=\frac{1}{4}\\ P(2)&=\frac{63}{216}=\frac{7}{24}\\ P(3)&=\frac{44}{216}=\frac{11}{54}\\ P(4)&=\frac{21}{216}=\frac{7}{72}\\ P(5)&=\frac{6}{216}=\frac{1}{36}\\ P(6)&=\frac{1}{216} \end{align}$$