How Is Probability Calculated When Choosing Balls of Different Colors?

[nhrock3]

in a jar we have 3 balls one red one white one black

3 people choose a ball and return it back.

x represents the number of different colors which the 3 people chose
y represents the number of white balls chosen.

p(x,y)
find p(2,0)

we have in total 27 possibilities
3 people chose 2 different colors(from the existing 3) and non of them was white

so each guy need to chose red or black out of 3 balls
$$\binom{3}{1}+\binom{3}{1}$$

why the answer is $$\frac{2\binom{3}{2}}{27}$$

 

[talolard]

You need the find all of the ways that 3 people can pick 3 balls out of the jar, in a way that no white ball was chosen.
How many options does each person have? does each person have the same number of options, given that the other person chose before him?