Number of possible outcomes

[trader10518]

Suppose there is a building with 18 windows per floor.
The architect says he would like to randomly select any 4 windows per floor to be toned in a different colour.
However, there should remain at least 2 regular windows between any pair of toned windows.

How can I calculate the number of possible combinations per floor?

Thanks,
Arthur

 

[joeu2004]

First, what you describe is a combinatorial math problem, not an Excel problem. Moreover, I suspect this is a class assignment. I'm sure you would not want us to spoonfeed a math solution, and rob you of the fun of developing it yourself (wink), based on your class instruction.

So if you provide a math formula for the solution, it is appropriate to ask this forum for help in implementing it in Excel -- unless your class is an Excel/VBA class.

Second, the problem statement is unclear. Does the rule about 2 plain windows between 2 toned windows apply only to each floor (horizontally), or also between floors (vertically!)?

For example, consider two possible 3-floor configurations (P=plain; T=toned):

(a)         (b)
PPTTPPTT    PPTTPPTT
PPTTPPTT    TTPPTTPP
PPTTPPTT    PPTTPPTT

Is #a allowed; or is #b required?

------

PS.... And is the building "circular"? In other words, consider the ends of the 18 windows on one floor (P=plain; T= toned):

(a)                       (b)
TTPP...10windows...PPTT    TTPP...8windows...PPTTPP

Is #a allowed; or is #b required because the 2 windows on the left after (considered) adjacent to the 2 windows on the right.

 

[joeu2004]

Second, the problem statement is unclear.

And finally, you write "randomly select any 4 windows". Must the 4 toned windows appear as 2 pairs (surrounded by 2 or more plain windows)?

Or is the following permitted, for example: TPTPTPT...11plainWindows.... ?

 

[joeu2004]

Oops! I exceeded the 10-min limit again! I just wanted to add....

My clarifying questions are for the benefits of others. I am not likely to particpate further, because combinatorial problems give me a headache (wink).

I would solve the problem by writing a VBA algorithm that exhaustively generates all 2^18 (262144) configurations for each floor (and possibly combinations of 2 and 3 floors), and count the "winners" based on the clarified rules.

But if we can consider each one floor independently, we could generate the 2^18 configurations in Excel cells, apply the clarified rules in another column, and count the "winners", all with Excel formulas.

Good luck!

 

[trader10518]

First, what you describe is a combinatorial math problem, not an Excel problem. Moreover, I suspect this is a class assignment. I'm sure you would not want us to spoonfeed a math solution, and rob you of the fun of developing it yourself (wink), based on your class instruction.

So if you provide a math formula for the solution, it is appropriate to ask this forum for help in implementing it in Excel -- unless your class is an Excel/VBA class.

Second, the problem statement is unclear. Does the rule about 2 plain windows between 2 toned windows apply only to each floor (horizontally), or also between floors (vertically!)?

For example, consider two possible 3-floor configurations (P=plain; T=toned):

(a)         (b)
PPTTPPTT    PPTTPPTT
PPTTPPTT    TTPPTTPP
PPTTPPTT    PPTTPPTT

Is #a allowed; or is #b required?

------

PS.... And is the building "circular"? In other words, consider the ends of the 18 windows on one floor (P=plain; T= toned):

(a)                       (b)
TTPP...10windows...PPTT    TTPP...8windows...PPTTPP

Is #a allowed; or is #b required because the 2 windows on the left after (considered) adjacent to the 2 windows on the right.

1. Well, it would be great to find a time machine to participate in class assignments again
2. Sorry for making the problem statement unclear:
- Building is not circular
- There is only one floor with 18 windows, just to simplify

"Must the 4 toned windows appear as 2 pairs (surrounded by 2 or more plain windows)?"

No, they must appear as single windows. And not necessarily surrounded, since a toned window may also be the first/last.

"Is the following permitted, for example: TPTPTPT...11plainWindows.... ?"

No, there is just one plain window between the toned. Should be a minimum of 2, like: TPPTPPT...10plainWindows...T

Here is the number of possible combinations:

But I have no idea how to subtract the ones which don't match the criteria to have at least 2 plain windows until the next toned.

Arthur