Regular expression to show that all strings contain each symbol atleast once
Your solution looks good to me, and it is probably what they expect of you.

It is interesting to consider the more general question: how large does a regular expression for this language be, as a function of the size of the alphabet? Denoting the size of the alphabet by $n$, Theorem 9 here shows a lower bound of $\Omega(c^n)$ for some (explicit) $c &gt; 1$. (The theorem is for context-free grammars, but a regular expression can be translated to a context-free grammar.) Your construction is $O(n\cdot n!) = 2^{O(n\log n)}$, so there is a some gap here.