Consider a function $f(x_1,...,x_n)$ of $n$ variables. How many different partial derivatives of order $r$ does $f$ possess?
The above problem was obtained from Sheldon Ross's "A first course in probability". I believe the answer should be $n^r$ partial derivatives, since there are $n$ choices for each independent variable as we differentiate successively from the $1$st to the $r$th derivative. However, I would like to verify this answer. I would also appreciate an alternative approach to the problem.
I found out that this problem is similar to the one solved here: How many n-th Order Partial Derivatives Exist for a Function of k Variables?
However, the accepted solution in the link above was based on the assumption that the order of differentiation does not matter. I believe this is only true when the given function is smooth (we weren't told it is). Even if we agree that the function is indeed smooth, isn't the answer supposed to be $n^r / r! $, given that we are considering the $r$th partial derivatives with distinct sets of independent variables only?