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BMT 2014 Spring - Discrete 6

Source:

January 6, 2022
probabilitycombinatorics

Problem Statement

Pick a 33-digit number abcabc, which contains no 00's. The probability that this is a winning number is 1a⋅1b⋅1c\frac1a\cdot\frac1b\cdot\frac1c. However, the BMT problem writer tries to balance out the chances for the numbers in the following ways:
[*] For the lowest digit nn in the number, he rolls an nn-sided die for each time that the digit appears, and gives the number 00 probability of winning if an nn is rolled. [*] For the largest digit mm in the number, he rolls an mm-sided die once and scales the probability of winning by that die roll.
If you choose optimally, what is the probability that your number is a winning number?
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