6. Among the 27 points consisting of the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of the cube, the number of groups of three collinear points is
A. 57
B. 49
C. 43
D. 37
Answer
As shown in the figure, the collinear triplets of points are divided into three categories. The first category: line segments connected by 8 vertices $C_{8}^{2}=28$; The second category: line segments connected by the midpoints of 12 edges $\frac{12 \times 3}{2}=18$; The third category: line segments connected by the centers of 6 faces 3. Therefore, the number of collinear triplets of points is $28+18+3=49$. So the answer is $B$.