Question
The number 12 may be factored into three positive integers in exactly eighteen ways, these factorizations include 1x3x4, 2x2x3 and 2x3x2.
Let N be the number of seconds in a day.
In how many ways can N be factored into three positive integers?
A general method is required, not just a numerical answer.
[This question is adapted from the UK MO for Girls 2011.]
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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.
I think it should be 12 combination 3
I thought that at first but I think it only works if all the items are indistinguishable, which they're not. You have to consider the fact that grouping something (x x y) (y y x) is slightly different from (y y y) (x x x) even though they both contain 3 items each.
It can be fracturised as many times as I think of elves in a day...lol
My brain fracturised at "fracturised" I think. I'll have to ask my bro. He's the math whizz in our family. But I'll take stock of the answer and try to sound smart next time someone talks about math to me :p