Question

As shown in the diagram, points $P_1,P_2,P_3,..........P_{10}$, are either the vertices or midpoints of the edges of a tetrahedran respectively. If the number of groups of four points

$(P_1,P_i,P_j,P_k)(1<i<j<k\le 10)$ lying on the same plane is ${}^{\prime }{m}^{\prime }$ then the sum of digits of ${}^{\prime }{m}^{\prime }$ is.

Answer 1

On each lateral face of the tetrahedron other than $P_1$ there are $5$ points.Take any $3$ points out of these $5$ and add $P_1$ to it. $(e.g.(P_1{P}_3{P}_4{P}_6\left)\right)$.
So, there are in all $3\times {}^5{C}_3$ required groups from lateral faces.
Apart from these, taking $3$ points on edge containing $P_1$ and another mid point from the edge which is not on the same plane with the edge considered above. We obtain another required group $(e.g.(P_1{P}_2{P}_{10}{P}_6\left)\right)$.
There are $3$ groups like this.
So, total number of required groups $m=33$
Sum of the digits in $m=6$