Determine all possible quintuplet(s) (A,B,C,D,E) of positive integers, with A ≤ B ≤ C ≤ D ≤ E, that satisfy this equation:

(A-1)*(B-2)*(C-3)*(D-4)*(E-5) = A+B+C+D+E

Prove that these are the only quintuplet(s) that exist.