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We are given a set of natural numbers 2, 3, 4, ..., n. Consider all subsets, each of them consisting of the combinations $_{(n-1)}C_{2}$, $_{(n-1)}C_{3}$, $_{(n-1)}C_{4}$ etc. We take the products of the terms in each such subset and then its reciprocals. Find the sum of all these reciprocals.

I believe I must find a recursive formula but I have no idea how to proceed...

I tried for n=4 and n=5 and found 5/12 and 86/120 respectively but I don't know how to continue. For example, for n=4: Let's consider the set {2,3,4}. Then we have the subsets {2,3}, (2,4), (3,4), (2,3,4) and their respective elements' products are 6, 8, 12 and 24. Then we take $\frac{1}{6}, \frac{1}{8}, \frac{1}{12}, \frac{1}{24}$ and add them. The result is $\frac{10}{24}$.

Also noticed that for each such fraction, the product of the numerator and the denominator equals n!.