Combinatorics of Street Lamp Arrangements

[kts1230]

Homework Statement

There are 17 street lamps along a straight street. In order to save electricity and not affect the regular use at the same time, we can shut down 5 of these lamps. But we cannot turn off a lamp at either end of the street, and we cannot turn off a lamp adjacent to a lamp that is already off. Under such conditions, in how many ways can we turn off 5 lamps?

Homework Equations

The Attempt at a Solution

I've looked at this question a few times now and I still don't even know where to begin. Any help would be highly appreciated.

 

[rude man]

Hmm .. I would start as follows:

First, forget about the two end-lights. They're always on. That leaves 15 lights to worry about.

How many ways can you arrange 15 distinguishable things taken 5 at a time?

Then, how many states are prohibited by the "no 2 adjacent lights off" rule?