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,…,n_k, the number of subsets satisfying the condition is

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