$ (T_n )$ = number of nonempty subsets of $ ({1, 2, 3, \dots , n})$ whose average is an integer. Call these subsets int-avg subset (just a name)
Note that one element subsets are by default int-avg subsets. They are n in number. Removing those elements from $(T_n)$ we are left with int-avg subsets with two or more element. We want to show that the number of such subsets is even.
Let X be the collection of all int-avg subsets S such that the average of S is contained in S
Y be the set of all int-avg subsets S such that the average of S is not contained in S.
Adding or deleting the average of a set to or from that set does not change the average.
This operation sets up a one-to-one correspondence between X and Y, so X and Y have the same cardinality. Since $latex (X\cap Y =\emptyset)$, the number of elements in $(X\cup Y)$ is even and hence the number of subsets of two or more elements that have an integer average is even.
Comment
What is the cardinality of $ (T_n)$?
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T_n - n is the number of non-singleton sets ......ar non- singleton sets always occur in pairs,er modhye there exists a S which contains the average.amr mne hoy this is a shorter solution.i was working it out.....