1. ## 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?

2. 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:
. . $\boxed{GG}\,, G, G, G, G, G, B, B, B, B, B, B,B,B$

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

. . . $\_\,B\,\_\,B\,\_\,B\,\_\,B\,\_\,B\,\_\,B\,\_\,B\,\ _\,B\,\_$

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

And there are: . ${8\choose6} \:=\:\boxed{84}$ ways.

3. 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?

4. That is a mere typo. It should be $\binom{9}{6}=84$
Nine blanks choosing six can be done in 84 ways.