Why does ways to pick $12$ elements from $5$ different types not equal to $5^{12}$? The question is how many ways you can pick $12$ donuts from $5$ different varieties. I thought I can just pick $1$ donut at a time. Each time I have $5$ choices for $1$ donut so it would be $5^{12}$ ways but it’s wrong.
 A: That would be correct if you were to compute the number of different line-ups of the donuts. However it is not the number of different bags of donuts, that is, disregarding the order in which they come.
A: "I thought I can just pick 1 donut at a time."
You can.  But that differentiates the order in which you pick them.  Picking a cherry glazed first and a buttermilk bar second will be different than picking a buttermilk bar first and a cherry glazed third.
So the number of ways to display twelve donuts in a row from $12$ different types is indeed $5^{12}$.
However it is assumed (perhaps frustrating in that they do not specifically state so) that order doesn't matter-- that you are just going to put the in a bag and you will eat them in any order you like--  That the customer is going to say "Gimme 4 glazed old fashioneds, 3 cherry glazed, and 5 buttermilk bars" and is not going to say "Oh and make sure you  put two of the glazed old fashioned between the second and third buttermilk bars and that none of the cherry glazeds are next to each other".
