From $3$ red, $4$ green and $5$ yellow balls, how many selections consisting of $6$ balls are possible, if each color must be represented twice?

Question

From $3$ red, $4$ green and $5$ yellow balls, how many selections consisting of $6$ balls are possible, if each color must be represented twice?

Answer 1

That question just makes sense if the balls of same color are distinguishable. If not, you have just 1 possible selection, taking 2 balls of each color.

So if they are distinguishable, you have $\binom{3}{2}$ options of choosing the red balls, $\binom{4}{2}$ of choosing the green, and $\binom{5}{2}$ of choosing the yellow. That are in total $3\cdot6\cdot10=180$ possible ways of choosing the balls.

Answer 2

Select two balls from each color; multiply the numbers of possibilities for each color together.