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?

\[ 5 ^ 2 \] \[ { 32 \choose 2 } \] \[ 3 ^ 5 - 1^5 \] \[ 2 ^ 5 \] \[ ^{32} P_2 \] \[ \frac{ 32 \choose 2 } { 2} \] \[ 3^5 - 2^5 \] \[ 3 ^ 5 \]

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