$32 = 2 ^ 5$ vertices of the form $( \pm 1, \pm 1, \pm 1, \pm 1, \pm 1)$ .

A hypercube in 5 dimensions hasFor each pair of distinct vertices, we connect them up with a line segment and mark the midpoint.

How many (distinct) midpoints have been marked?

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Number of vertices in a n-hypercube $={ 2 }^{ n }$

Number of lattice points in a n-hypercube including all mipoints $={ 3 }^{ n }$

Number of midpoints $={ 3 }^{ n }-{ 2 }^{ n }$