How many ways are there to put 8 beads of different colors on the vertices of a cube, if rotations of the cube (but not reflections) are considered the same?

Actually, this is the correct answer.

Consider one vertex of the cube. When the cube is rotated, there are 8 vertices which that vertex could end up at. At each of those vertices, there are 3 ways to rotate the cube onto itself with that vertex fixed. So, there are a total of 8*3=24 ways to rotate a cube. There are 8! ways to arrange the beads, not considering rotations. Since the arrangements come in groups of 24 equivalent arrangements, the actual number of ways to arrange the beads is 8!/24=1680.

12

step-by-step explanation:

both the histogram and the stem and leaf plot will visually show how many shoppers were of various ages within each range of data.   the histogram does not give us the specific ages of these shoppers unlike the stem and leaf plot.a histogram does not give us a true range of the data because we do not know exactly how old the shoppers were. a stem and leaf plot will give exact ages, so we can determine the exact range along with all measures of center and the others measures of center.   with the histogram, we cannot find any of these.

step-by-step explanation: