An edge set and the number of triangles in $K_n$

Question

Let $E' \subseteq E(K_n)$ and $T_i = \{H\subseteq K_n \equiv K_3 : |E' \cap E(H)| = i\}$, (i.e. the triangles in $K_n$ such that $E'$ contains exactly $i$ edges of that triangle).
Show that $|T_1| + |T_2| < n^3/8$. I was thinking this quantity was $n/2 * n^2/4$ and the max number of edges of a triangle free graph is $n^2/4$.