My question is that in how many ways can $10,000!$ be written as the product of $30$ distinct positive integers. My question is similar to this question: In how many ways can $1000000$ be expressed as a product of five distinct positive integers? but mine is much larger.
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There are $K=1229$ primes $\leq N:=10,000$. For each such prime $p$ compute $m=\lfloor N/p\rfloor+\lfloor N/p^2\rfloor+\ldots$ (a finite sum). This $m$ is the exponent of $p$ in $N!$. In this way you get an array ${\bf m}=(m_k){1\leq k\leq K}$. Now you have to count solutions of $\sum{i=1}^{30}{\bf x}_i ={\bf m}$ satisfying certain constraints. – Christian Blatter May 12 '16 at 09:02
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Why the above should not be a question about mathematics is beyond me. It's again so that certain "Oberlehrers" didn't see a (maybe approximate) solution right away; so they decided to close the question. – Christian Blatter May 12 '16 at 09:07
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@ChristianBlatter Hmmm. So would the only way to calculate this be with a computer program? If so, I'm not used to programming, how would I program this? – user338943 May 13 '16 at 03:38
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@ChristianBlatter Can we calculate this using wolfram alpha? Can you give a link if possible? I'm not very familiar with stars and bars and its application to this problem – user338943 May 13 '16 at 14:43