I noticed a building outside my house with randomly lit rooms and dark rooms. If I treat each one of those windows as a square, is it possible to calculate the total number of patterns that can be created by illuminated rooms. Say there are 25 windows in total. (5*5) I don't know if I'm being articulate enough with my problem.
I tried using Combinations, but I'm not sure how to approach it properly.
In a 2*2 matrix, the number of patterns that can be created is 12.(Including the pattern where none of the rooms are lit). I calculated this without using a formula, just by logic. Is there a simple method I can use to find out a similar answer to a 5*5 square, without using a computer simulation?
If computer simulations are to be used, how would one go about solving this, and what is the principle involved in this kind of simulation?
Here's how people get their formulae, with an example,the following is a set of things:
$$\{a,b,c,d,e\}$$
Lets use exponents to represent the value of the state a given element is in (two states in your example for each, lets do the same for mine using 0 and 1), we then have the following combinations: $$\begin{equation*}a^0b^0c^0d^0e^0\\a^0b^0c^0d^0e^1\\a^0b^0c^0d^1e^0\\a^0b^0c^0d^1e^1\\a^0b^0c^1d^0e^0\\a^0b^0c^1d^0e^1\\a^0b^0c^1d^1e^0\\a^0b^0c^1d^1e^1\\a^0b^1c^0d^0e^0\\a^0b^1c^0d^0e^1\\a^0b^1c^0d^1e^0\\a^0b^1c^0d^1e^1\\a^0b^1c^1d^0e^0\\a^0b^1c^1d^0e^1\\a^0b^1c^1d^1e^0\\a^0b^1c^1d^1e^1\\a^1b^0c^0d^0e^0\\a^1b^0c^0d^0e^1\\a^1b^0c^0d^1e^0\\a^1b^0c^0d^1e^1\\a^1b^0c^1d^0e^0\\a^1b^0c^1d^0e^1\\a^1b^0c^1d^1e^0\\a^1b^0c^1d^1e^1\\a^1b^1c^0d^0e^0\\a^1b^1c^0d^0e^1\\a^1b^1c^0d^1e^0\\a^1b^1c^0d^1e^1\\a^1b^1c^1d^0e^0\\a^1b^1c^1d^0e^1\\a^1b^1c^1d^1e^0\\a^1b^1c^1d^1e^1\end{equation*}$$
Which has $32=2^5$ states in your case you have $2^{25}=33,554,432$ states, assuming rotations, reflections, and translations aren't considered duplicates.