Another theatre line problem

[MickeyHoldem]

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Ok, I think I went about this the hard way and had to look up a sequence to get the answer.

So I computed the answers for N = 1, 2, 3, and 4.

N = 1, I got 1 / 1.
N = 2, I got 3 / 2.
N = 3, I got 11 / 6.
N = 4, I got 50 / 24.

So I have the denominators as factorials, and the numerator was this sequence that I didn't know off the top of my head. So, I looked up the sequence 1, 3, 11, 50. And I found that this is the sequence of Stirling Numbers (of the 1st kind)... some important combinatorial sequence that I vaguely learning a little bit about years ago.

So, I looked up the 10th Stirling number (indexed from 1), which was 10628640. And, 10! = 3628800.

So, we have 10628640 / 3628800 = 2.929.

-RMJ

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I went the same way as you.... I had never heard of Stirling numbers. I went to N=5 for 274/120 and then found the relationship with the Harmonic series when I was trying to figure out the numerators. The numerator (Stirling Number) is just Harmonic Number(n) * n!