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.
Let a=A-1 b=B-2.... e=E-5
Let P= abcde and S=A+B+...E
Considering the range and the ascending order constraints, and starting with (1,1,1,1,p),then (1,1,1m,p) etc one quickly arrives to 28 and 40 as the only solutions of P=S.
Now permuting (1,1,2,2, 7) - 6 combinations with 7 as last number and permuting 1,1,1,2, 20 - 4 combinations with 20 as last number -yields the 1O sets of (abcde) from which the corresponding sets of (ABCDE) are derived.