Problem on number of amino acids that an alien can use
I also got 624. Maybe the answer is incorrect.
My processing is:

Each codon has four bases, the species contain five unique nucleotides, so for the first position in the codon, there are 5 possible results, also, for the second position in the codon, there are 5 possible results, it is same in third and forth position,

so the result should be  5*5*5*5-1=624