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If the 12 signs end as "a" signs together first, followed by "b" signs together, followed by "c" signs together, and so on, let's write it as a/b/c....
We could assume a≤b≤c... if we also take into account in how many different ways we can permute the numbers a, b, c....
The possible ways to get a/b/c... are 12!/a!b!c!... multiplied by the number of ways to order a, b, c...
For example, with 1/2/2/7, the number is 12!/1!2!2!7! times 4!/1!2!1!.
Summing over the 70 possible combinations of a/b/c..., we get the final answer: 28,091,567,595.
Note: Looking around in the web, I found the first problem (with 8 horses instead of 12 signs) and a different way of solving it at "Nick's Mathematical Puzzles". |
