SOLUTION: This is in the binomal theorem section. Suppose that a set has an odd number of elements. Explain why half of the subsets will have an odd number of elements.

The binomial expansion (x+y)n has n+1 terms

So if n is odd then (x+y)n has an even
number of terms.

A set with n elements has 2n subsets.

There are nCk subsets with k elements.

2n = (1+1)2 has an even number of
terms, and each term is nCk, the number of subsets with
k elements.
 
since nC0=nCn, nC1 = nC(n-1), nC2 = nC(n-2),...,nC(2n-1)/2=nC(2n+1)/2

Every term in the first half of the terms equals a term
in the second half of the terms.

That's why  half of the subsets will have an odd number 
of elements.

That isn't true of sets with an even number of elements,
because (1+1)n has an odd number of terms when
n is even.

Edwin