Non-Splittable Subsets

Probability Level 5

Let SS be a set of three integers chosen from N={1,2,,2012}N = \{1,2,\ldots,2012\}. We say that SS is 4-splittable if there is an nNSn \in N \setminus S such that S{n}S \cup \{n\} can be partitioned into 2 sets such that the sum of each set is the same. How many 3-element subsets of NN are not 4-splittable?

Details and assumptions

You can refer to Set Notation for the definition of NS N \setminus S.

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