In how many ways can seven beads of distinct colors be put on the hexagonal grid shown, if reflections and rotations of an arrangement are considered equivalent?
Second one -----
A row of 8 light switches controls different banks of lights in a school gymnasium. In preparation for a school dance, students decide that they want either one or two banks of lights on during the dance. How many possible ways are there of setting these 8 switches so that either one or two banks of lights are on?
7 ways to chosse the middle one
Put another down anywhere
5! ways to order the rest
So that is 840
that takes care of rotation but not relfections
There are six axes of symmetry so I'll have to divide by something...not sure what.
2 of the places are fixed before the other 5 get included so maybe there is only really one reflection that counts.
840/2 = 420 But that is a bit of a guess but is seems right .... well maybe anyway.
I do not understand the second question - I do not know what is being asked.