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4 Let N be an integer greater than 1 and let $(T_n)$ be the number of non empty subsets S of ({1,2,…..,n}) with the property that the average of the elements of S is an integer.Prove that $(T_n - n)$ is always even.

Sketch of the Proof:

$(T_n )$ = 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 $(X\cap Y =\emptyset)$, 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 $(T_n)$?