The points $P_1, P_2, \dots, P_{10}$ are either vertices or midpoints of edges of a tetrahedron. For $1 < i < j < k \le 10$, how many groups of four coplanar points $(P_1, P_i, P_j, P_k)$ exist? (Where $P_1$ is a vertex.)
The number of points which lie on each face as a plane are $3\binom{5}{3}$ but the answer is given as $3\binom{5}{3} + 3$.