Interpreting combinatorics: how are these not the same question? Here are two allegedly different questions from Lovasz's Combinatorial Problems and Exercises:


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*In a shop there are $k$ kinds of postcards. We want to send postcards to $n$ friends. How many ways can it be done?





*We have $k$ distinct postcards and want to send them all to our $n$ friends (a friend can get any number of them, including 0). How many ways can it be done?


These appear to me to be the same question. How are they different?
 A: The two problems describe different situations. In particular, in problem $2$ we have $k$ distinct postcards whereas in problem $1$ there are $k$ types of postcards. Moreover, it appears that the intended number of postcards per friend is different, albeit this is not stated explicitly.
Problem $2$
Let's begin with problem $2$ since it is stated more clearly. We have $k$ distinct postcards. Obviously, when we write the name and address of the recipient on the first postcard we choose from among our $n$ friends. Now, the key observation is that when we write the name and address of the recipient on the postcard for any other friend we still choose from among the $n$ friends since - as is explicitly clarified - a friend can get any number of postcards. Thus, by the time we are done with all $k$ postcards we will have made $k$ independent choices each from among $n$ options. Therefore, there are $n^k$ ways we can address the postcards in problem $2$.
Problem $1$
Problem $1$ is less explicit. First, it is not explicitly stated whether we intend to send a single postcard to each friend. However, sending a single postcard to each friend is what people do typically. Moreover, in the second problem we do send fewer or more postcards to each friend and this is stated explicitly. Therefore, we assume that each friend gets one postcard. Second, we don't know how many postcards of each type the shop has. Since the inventory is unspecified we will assume that it is effectively limitless.
One postcard per friend rule calls for a procedure different than in problem $2$. Namely, here we walk through the list of $n$ friends and for each we choose which of the $k$ types of postcard to send to that friend. Since the inventory is unlimited the choices are independent. Therefore, we make $n$ independent choices each from among $k$ options. Consequently, there are $k^n$ ways we can choose postcards for our friends in problem $1$.
