Number of arrangements of noughts and crosses in grid 
The way I went about it is as follows: 


*

*let's consider a 4x4 grid with 15 noughts and 1 cross. This would generate 16 different combinations.

*Now for a 4x4 grid with 4 crosses and 12 noughts, each cross could take 12 different positions and since we have 4 crosses then the number of possible arrangements is 4*12=48. 


Is this right or is there a different way of approaching this kind of questions. 
Thanks 
 A: If each of the $4$ crosses would have $12$ possibilities, it would be more like $12*12*12*12$, i.e. $12^4$
But that is not right: the first cross has $16$ possibilities, the next one $15$, the next $14$, and the next $13$ ... so that suggests $16*15*14*13$ ...
... but that is not right either!!  
Why?  
Well, putting the first cross in the top right corner, and the second cross in the top left corner will give the same outcome as when you put the first cross in the top right corner, and the second cross in the top left corner. So, the $16*15*14*13$ formula is overcounting the number of possibilities! 
In fact, if we distinguish between the crosses, then there are $4!$ ways to put the $4$ crosses in $4$ positions, and in the $16*15*14*13$ formula we treat each of those ways as different (i.e. the formula acts as if all the crosses are different).  But since the crosses are all the same, we end up counting each configuration $4!$ times, so we should divide $16*15*14*13$ by $4!$.
Here is how to conceptually think about this: Out of $16$ possible positions, you need to pick $4$ ... that is exactly what the $16 \choose 4$ formula is, and not coincidentally:
$${16 \choose 4} = \frac{16!}{12!4!} = \frac{16*15*14*13}{4!}$$
