Lets say you have n x n grid, and each square on the grid is either black or white. How many different combinations of this grid can exist?

I figured 9! would be for the grid, but I don't know how to incorporate the fact each tile can be 2 different things.

  • $\begingroup$ Is there anything other than black or white going on? Like numbered squares or something? $\endgroup$ – Glen_b -Reinstate Monica Sep 13 '17 at 22:49
  • $\begingroup$ nope there isnt $\endgroup$ – Itachi San Sep 13 '17 at 22:50
  • $\begingroup$ 1. Where did the "9" come from? 2. Each square can be in two states. Start with one square. Two possibilities (B or W). Add a square (BB, BW,WB,WW) etc. Every new square simply doubles how many possibilities there are. $\endgroup$ – Glen_b -Reinstate Monica Sep 13 '17 at 22:52
  • $\begingroup$ @Glen_b I think the OP meant 9 factorial. It still wouldn't make sense. $\endgroup$ – Michael R. Chernick Sep 14 '17 at 0:43
  • $\begingroup$ @Michael Yes thanks, I saw the "!", but the question is what any specific number would be doing in a question about $n$. $\endgroup$ – Glen_b -Reinstate Monica Sep 14 '17 at 1:00

Each square can have one of two possible values. There are $n^2$ squares. Therefore the number of possibilities is $2^{n^2}$


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