A window is 2 by 8 ft. it consists of 8 panes each measuring 2 by1 ft. How many ways can the window panes be arranged

Question

a window is 2 by 8 feet. It consists of 8 window panes each measuring 2 by1 feet how many different arrangements can the panes be put in if the panes cannot be cut

Answer 1

Often, a picture really helps:

The window:

  XXXXXXXX

  XXXXXXXX

 

A pane:

   HH          [horizontal]

 or

   V            [vertical]

   V

 

Constraint:

   8 panes of 2 ft * 1 ft = 16 sq-ft;  yes, 2 sq-ft * 8 panes = 16 sq-ft, so the window will be “full”

 

Consideration:

  What makes a unique “way” of arranging panes?  For example, if one arrangement is the “mirror image” (flip right-to-left) of another arrangement, is that a different arrangement?

 

Let’s assume that each “way” of arranging panes counts for the “how many?”  (this makes the problem simpler)

 

Note that “sliding” the position of the panes is possible.  Here is how one such case would look:

   V  HH  HH  HH  V

   V  HH  HH  HH  V

 

Thinking about that picture, how many ways can V and H be place into a row totaling 8 if V=1 and H=2?

 

Each of the following lists one of the duplicate rows in a solution:

 1.      V V V V V V V V        [all vertical]

 

 2.      V V V V V V H            [one horizontal set in various positions)

 3.      V V V V V H V

 4.      V V V V H V V

 5.      V V V H V V V

 6.      V V H V V V V

 7.      V H V V V V V

 8.      H V V V V V V

 

 9.      V V V V H H            [two horizontal sets in various positions]

10.     V V V H V H

11.     V V V H H V

12.     V V H V V H

13.     V V H V H V

14.     V V H H V V

15.     V H V V V H

16.     V H V V H V

17.     V H V H V V

18.     V H H V V V

19.     H V V V V H

20.     H V V V H V

21.     H V V H V V

22.     H V H V V V

23.     H H V V V V

 

24.     V V H H H          [three horizontal sets in various positions]

25.     V H V H H

26.     V H H V H

27.     V H H H V

28.     H V H V H

29.     H V H H V

30.     H H V V H

31.     H H V H V

32,     H H H V V

 

33.     H H H H         [four horizontal sets in each row; eight panes total]

 

There are 33 possible arrangements.    It is now easy to eliminate "mirror-image" duplicates (e.g., (2) and (8)) if that is the assumption.