I'm having trouble with this question about regular expressions:
Let $L_1$ be the language of passwords containing at least one lowercase letter, at least one uppercase letter, and at least one number. Write down a regular expression that describes $L_1$.
This is my answer, but I'm not too sure about it. I let $\mathcal{A}$ be the alphabet, and attempted to cover all permutations. I think my answer is correct, but is there a simpler way to go about it? \begin{equation*} (\mathcal{A}^*\mathcal{A}_{LC}\mathcal{A}^*\mathcal{A}_{UC}\mathcal{A}^*\mathcal{A}_{No}\mathcal{A}^*) \cup (\mathcal{A}^*\mathcal{A}_{LC}\mathcal{A}^*\mathcal{A}_{No}\mathcal{A}^*\mathcal{A}_{UC}\mathcal{A}^*) \cup (\mathcal{A}^*\mathcal{A}_{No}\mathcal{A}^*\mathcal{A}_{LC}\mathcal{A}^*\mathcal{A}_{UC}\mathcal{A}^*) \cup (\mathcal{A}^*\mathcal{A}_{No}\mathcal{A}^*\mathcal{A}_{UC}\mathcal{A}^*\mathcal{A}_{LC}\mathcal{A}^*) \cup (\mathcal{A}^*\mathcal{A}_{UC}\mathcal{A}^*\mathcal{A}_{LC}\mathcal{A}^*\mathcal{A}_{No}\mathcal{A}^*)\cup (\mathcal{A}^*\mathcal{A}_{UC}\mathcal{A}^*\mathcal{A}_{No}\mathcal{A}^*\mathcal{A}_{LC}\mathcal{A}^*) \end{equation*}
I'm really stuck on this second part:
Let $L_2$ be the language of "good" password. It is related to L1 by: \begin{equation*} L_2=\{w\in L_1 : w \text{ has length } \geq 8\} \end{equation*} Specify a regular expression that describes $L_2$, without writing it down in full. What is the length of this regex?
I'm not sure about how to construct a regular expression using my answer for $L_1$. I thought it might be similar to the first part, where I could specify all the permutations.