Le Monde puzzle [#822]

For once Le Monde math puzzle is much more easily solved on a piece of paper than in R, even in a plane from Roma:

Given a partition of the set {1,…,N} in k groups, one considers the collection of all subsets of  the set {1,…,N} containing at least one element from each group. Show that the size of the collection cannot be 50.

Obviously, one could consider a range of possible N’s and k’s and run a program evaluating the sizes of the corresponding collections. However, if the k groups are of size n1,…,nk, the number of subsets satisfying the condition is

(2^{n_1}-1)\times \ldots \times (2^{n_k}-1)

and it is easily shown by induction that this number is necessarily odd, hence the impossible 50.

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