Footed Glass

Maths Question - Please Help!?

I'm a year 9 student, just got stuck on this question for homework. Could you please explain and complete this question with full working to show me how you did it? Thanks !

Problem:
A contractor is building new houses. Each house has a large window with width 2 feet and height 8 feet to be made up of eight identical 1 foot by 2 feet glass panes.

The contractor wants each house to look slightly different. The window size is fixed, but he has plans on arranging the the glass panes in the windows so that no two windows have the same configuration.

Assuming the glass panes cannot be cut, how many different arrangements can be made?

Any help would be much appreciated, thanks!

I'll draw this picture sideways - it's more convenient.

_ _ _ _ ← 8 → _ _ _
| . . . . . . . . . . . . . . . .|
| . . . . . . . . . . . . . . . .|. ↑
- - - - - - - - - - - - - - - - | 2
| . . . . . . . . . . . . . . . .|. ↓
| _ _ _ _ _ _ _ _ _ _ |

Start at the left:
He can put a 1x2 vertically, or 2 of them horizontally.

If he puts one vertically, he has 7 feet left to work in,
and if he puts 2 horizontally, he has 6 feet left to work in.

So F(8) = F(6) + F(7) (where F(n) = number of ways to "do" n feet in this fashion)

Stop here if you want to think about solving this yourself.

Similarly, for smaller numbers F(n) = F(n-1) + F(n-2)
which is the familiar formula for the marvelous sequence known as the Fibonacci numbers.

F(1) = 1 way (a single vertical pane)
F(2) = 2 ways (horizontal or vertical)
and then you have it:
F(3) = 1 + 2 = 3
F(4) = 3 + 2 = 5
F(5) = 5 + 3 = 8
F(6) = 13
F(7) = 21
and finally
F(8) = 34

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