ABC is a three digit number in which the digit A is greater than the digits B and C. If the difference between ABC and CBA is 297 and the difference between ABC and BAC is 450, find all possible three digit numbers ABC and find their sum also.

Writing $$ABC-CBA=297$$ gives $$A-C=3$$, and $$ABC-BAC=450$$ gives $$A-B=5$$, so combining these gives $$C-B=2$$. Testing valid digit values with $$A>B$$ and $$A>C$$ from $$A=9$$ downward gives the four valid numbers 946, 835, 724 and 613 (with $$A=5$$ giving an invalid case since B would be 0, violating the non-zero digit condition implicitly assumed, or otherwise not forming a valid distinct case). Adding these gives $$946+835+724+613 = 3118$$.