Can't we just use Stars and Bars here? There are 86400 seconds in a day and 86400 = 27 • 33 • 52.
According the S&B Theorem, there are (n+k-1)C(k-1) ways to divide up n items into 3 bins (with the caveat that you're allowed to place no items in a bin - the results are different if you aren't allowed to have a factor of 1). In this case, our "bins" might be indicated by parentheses around certain groups of numbers. n is the number of factors of a certain prime number, and k always equals 3 because there are 3 bins.So, there are:
9C2 ways to place the seven 2's,
5C2 ways to place the three 3's
4C2 ways to place the two 5's
Since all three events of placing different factors are independent, we multiply them together, and this gives 36 x 10 x 6 = 2160.
Can't help thinking that there's more to it than this, though.
Yes, not so hard when one knows how! I do a lot of combinatorics with my gifted students because such questions often appear in competitions but the topic is sadly lacking in most school maths. The students also can get mightily tangled up so we have to revert to "playing the game" by hand till the formulas make sense.
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