[1 Problem](https://artofproblemsolving.com#Problem)

[1.1 Solution 1](https://artofproblemsolving.com#Solution_1)
[1.2 Solution 2](https://artofproblemsolving.com#Solution_2)


[2 See also](https://artofproblemsolving.com#See_also)


Solution 1
There are $8$ choices for the first endpoint of the line segment, and $7$ choices for the second endpoint, giving a total of $8\cdot 7 = 56$ segments.  However, both $\overline{AB}$ and $\overline{BA}$ were counted, while they really are the same line segment.  Every segment got double counted in a similar manner, so there are really $\frac{56}{2} = 28$ line segments, and the answer is $\boxed{D}$.
In shorthand notation, we're choosing $2$ endpoints from a set of $8$ endpoints, and the answer is $\binom{8}{2} = \frac{8!}{6!2!} = 28$.

Solution 2
Each segment is either an edge, a facial diagonal, or a long/main/spacial diagonal.
A cube has $12$ edges:  Four on the top face, four on the bottom face, and four that connect the top face to the bottom face.
A cube has $6$ square faces, each of which has $2$ facial diagonals, for a total of $6\cdot 2 = 12$.
A cube has $4$ spacial diagonals:  each diagonal goes from one of the bottom vertices to the "opposite" top vertex.
Thus, there are $12 + 12 + 4 = 28$ segments, and the answer is $\boxed{D}$.
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