Count of subsets, that consist of 4 elements where the sum is even

Question

Let $n \ge 4$.

$G_4$(n) is the total count of subsets, that consist of four elements $ A \subseteq$ {1,...n}, where the sum of elements of A is even.

How can I count $G_4$(n)?

I think I can calculate it with the binomial coefficient, where k = 4, but what would my n be?

Answer 1

Hint

The sum of four elements will be even, if there is even ammount of odd elements.

If it's not enough...

Split the set into two subsets - one contains only even numbers ($B$), and second - only odd numbers ($C$).

We have two cases:

1. $n=2k$. In this case we have: $|B|=|C|=\frac{n}{2}$
2. $n=2k+1$. In this case we have: $|B|=\frac{n-1}{2}$, $|C|=\frac{n+1}{2}$

Now, for $n\leq 6$ we can select elements in: $$G_4(n)=\binom{|B|}{2}\binom{|C|}{2}$$

For $n=7$ (at least 4 odd numbers, but only 3 even numbers): $$G_4(n)=\binom{|B|}{2}\binom{|C|}{2}+\binom{|C|}{4}$$

And for $n>7$ (at least 4 odd and 4 even numbers): $$G_4(n)=\binom{|B|}{4}+\binom{|B|}{2}\binom{|C|}{2}+\binom{|C|}{4}$$