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3)How mnay subsets of 6 integers taken from the numbers 1,2,3...,20 are
there such that there are no consecutive integers in any subset (e.g. if 5 is in the subset then 4 and 6 cannot be in it)?

This is a fairly challenging problem, what have you tried so far?
There are 20 choose 6 subsets, so I doubt you could use trial and error here.

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