5-D on 2-D

Geometry Level 5

A hypercube in 5 dimensions has $32 = 2 ^ 5$ vertices of the form $( \pm 1, \pm 1, \pm 1, \pm 1, \pm 1)$ .

For each pair of distinct vertices, we connect them up with a line segment and mark the midpoint.

How many (distinct) midpoints have been marked?

2 ^ 5

3 ^ 5

\frac{ 32 \choose 2 } { 2}

^{32} P_2

3 ^ 5 - 1^5

5 ^ 2

{ 32 \choose 2 }

3^5 - 2^5