Let $\Sigma$ be the Latin alphabet ({a, b, c, ..., x, y, z} - 26 letters).
Given the language L = {$\alpha \in \Sigma$*| if $\alpha$ has the letter a, then $N_a(\alpha)$ = 4 and if $\alpha$ has the letter b then $N_b(\alpha)$ = 8 and ... if $\alpha$ has the letter z then $n_z(\alpha) = 2^{27}$ }, prove that L is regular.
How exactly should I go about writing a formal proof on this? I tried to play around with $a^{-1}L$ but failed to accomplish much.
Any help is deeply appreciated.
Note that the language is finite: "a" may only appear $0$ or $4$ times, "b" $0$ or $8$ and so on. For each of the $2^{26}$ possible choices of the counts of letters in $\alpha$ there are only a finite number of ways to permute them, leading to a finite number of words in $L$ for the choice made. All finite languages may be recognised by a finite-state machine; finite-state machines recognise precisely the regular languages.