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I have to compute $|\{(a,b,c)\in \mathbb{N}*\mathbb{N}*\mathbb{N}: abc=5 000 000\}|$. My result is $56*7!*8!$, but I'm not sure of this.

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We have to distribute independently $6$ indistinguable twos and $7$ indistinguable fives into the three labeled boxes $a$, $b$, $c$. According to stars and bars this can be done in $${8\choose 2}\cdot{9\choose2}=1008$$ ways.

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    I agree with the general plan, but how does stars-and-bars give you $\binom 72 \binom 82$ rather than $\binom{6+3-1}{3-1}\binom{7+3-1}{3-1} = \binom 82\binom 92$? – 2017-01-23
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Assuming that the order of $(a,b,c)$ is significant.

$5000000=2^6\cdot5^7$.

Hence we choose:

  • $a=2^x\cdot5^y$
  • $b=2^z\cdot5^w$
  • $c=2^{6-x-z}\cdot5^{7-y-w}$

For $x\in[0,6]$ we have $7$ options.

For $y\in[0,7]$ we have $8$ options.

For $z\in[0,6-x]$ we have $7-x$ options.

For $w\in[0,7-y]$ we have $8-y$ options.

Hence the answer is $\left(\sum\limits_{x=0}^{7}\sum\limits_{z=0}^{6-x}1\right)\cdot\left(\sum\limits_{y=0}^{7}\sum\limits_{w=0}^{7-y}1\right)=28\cdot36=1008$.

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    Hmmm... A little unsure of this answer, let's wait for some feedback from the community... – 2017-01-23
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    Looks right to me. Checked with the code from https://oeis.org/A007425. – 2017-01-23