From the total pool of colored balls, one can choose 2 reds, 2 blacks, 3 whites, and 2 blues in
ways.
I'm assuming no ball of the same color is distinguishable from any other ball of the same color. So when I'm considering the possible arrangements, if I had lined up the ball as
red1 - black - red2 - ...
then this would be no different that
red2 - black - red1 - ...
So I now have 9 balls to arrange, which means there are
total possible permutations of them. But order among distinct colors is assumed to not matter. This means I have to divide the total number of permutations by the number of ways I could permute balls of the same color. Then there would be a total of
ways of arranging the balls I had selected.
ways.
I'm assuming no ball of the same color is distinguishable from any other ball of the same color. So when I'm considering the possible arrangements, if I had lined up the ball as
red1 - black - red2 - ...
then this would be no different that
red2 - black - red1 - ...
So I now have 9 balls to arrange, which means there are
ways of arranging the balls I had selected.