KöMaL 2009 K199

Dixie puts five-digit numbers in order, according to the following rule: First he arranges them in decreasing order of the last digit. If the last digit is the same in two numbers, the one with the smaller first digit will precede the other. If their first digits are also the same, they will be listed in decreasing order of the product of the middle three digits. (Dixie only deals with numbers whose order can be decided by this rule.) Dixie wrote six numbers on a sheet of paper. The numbers were ordered according to the rule above, but some digits got blurred. (The blurred digits are replaced by letters here.) It reads: 42348, A8318, 56B48, 8653C, 46585, D8655. Considering all possibilities, we try to guess the six numbers. (Our guess is one possible value for each number.) What is the probability that we will get all Dixie's numbers right?