Another theatre line problem

[RocketManJames]

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