A $4$-storey house is to be painted by some $6$ different colors such that each storey is painted in one color. How many ways are there to paint the house? Repetition of color is allowed.
The answer turns out to be $6^4$. Although this seems correct to me if looking at the problem color by color I was wondering why can't the answer be $4^6$ instead looking at the problem floor by floor?
On
Well, if you are to paint one of the storey of the $4$ - storeyed house, you have $6$ possible choices. And you paint the first storey AND the second one AND the third one AND the fourth one where for each of them, you hae 6 choices. So, total choices in which you can paint your $4$ - storey house is $6^4$.
Using $6^4$ means that the color repetition is allowed but the floor repetition is not allowed, while using $4^6$ means that the floor repetition is allowed (contradicts the fact that each floor can only be colored once) but the repetition of colors is not allowed.