# Marbles of different colors in a row.

We have 10 red marbles, 6 green and 5 blue (marbles of the same color are identical) and want to arrange them in a row in such a way that there are not any consecutive green marbles. In how many ways is this possible?

We can place the 6 green marbles in a row - this leaves 7 gaps in between them plus beginning + end. Then we must distribute the 10 red and 5 blue marbles in such a way that none of the middle gaps (empty boxes) remains empty.

Alternatively, we can place the 10 red and 5 blue marbles in a row, with an empty space between any two balls, regardless of the color: this leaves 16 gaps.

Then we must place 6 marbles in these 16 gaps. This is C(16,6).

For the 10 red and 5 blue marbles, there are $$\frac{15!}{10!5!}$$ ways to distribute them.

So the total way is the multiplication of these two?

Am I right?

Thank you in anticipation!

• Yes, you're exactly right. I prefer the second strategy for the \$myself -- there are people here who make the first one work, but I think the second way is more elegant for the passive reader. – Matthew Daly Aug 21 at 9:41