Thread: Permutations of Bottles

In how many different ways can I arrange 7 green and 8 brown bottles so that exactly one pair of green bottles are side by side?

I've been trying to do this all day, am I right for separating the answers into 14 cases of where the 2 green bottles can be across 15 places?

Hello, mixtapevanity!

In how many different ways can I arrange 7 green and 8 brown bottles
so that exactly one pair of green bottles are side by side?

Duct-tape two green bottles together.

Now we have 14 units to arrange:
. .

Now place the eight brown bottles in a row.
. . Note that there are spaces before, after and between them.

. . .


We will take the six green units and place them in six of the nine spaces.

And there are: . ways.

Thanks for your reply! That is along the lines of what I was thinking. But can I ask what did you mean by (8 6) = 84?

That is a mere typo. It should be
Nine blanks choosing six can be done in 84 ways.