Hello, mixtapevanity!

Duct-tape two green bottles together.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?

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.