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 \((T_n)\)?

*Related*

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…..