Given is a set S = {A, C, G, T}. We are interestend in the numbers of elements in the biggest subset from S^n (n natural number), which fulfills the following restriction: The amount of "G" and "C" together occuring in an Sequence is n\2 for even n, and (n-2)/2 for uneven n.
Example 1: n = 6, even. {C,G} has to occur 3 times. 2 of the acceptet sequences are: (C,A,G,C,T,A) und (G,G,G,A,A,A) 2 of the NOT acceptet sequences are: (A,G,T,A,T,C) und (C,C,C,C,A,A)
Beispiel2 n = 3, not even, {C,G} has to occur 1 time. 2 of the acceptet sequences are: (C,A,T),(A,A,G) 2 of the NOT acceptet sequences are: (C,G,T),(T,A,A)
Almost half of them will be filled from ${C,G}$. Specifically in a sequence of $n$ elements, $\lfloor{n \over 2}\rfloor$ will be filled from ${C,G}$ and $\lceil{n \over 2}\rceil$ will be filled from ${A,T}$. Since we should choose $\lfloor{n \over 2}\rfloor$ positions from $n$ positions to be filled from ${C,G}$ we get
$$ S_n = {n \choose \lfloor{n \over 2}\rfloor}2^{\lfloor{n \over 2}\rfloor}2^{\lceil{n \over 2}\rceil} = {n \choose \lfloor{n \over 2}\rfloor}2^n $$
(Hope, I understood your question)