### Instruction: How many ways are there to put 9 differently colored beads on a $3\times3$ grid if the purple bead and the green bead cannot be adjacent (either horizontally, vertically, or diagonally), and rotations and reflections of the grid are considered the same?### Response: I want to count the number of arrangements of 9 beads on a $3\times3$ grid, but I have to avoid the cases where the purple and green beads are adjacent.
One way to do this is to fix the position of the purple bead, and then count the number of ways to place the green bead and the other 7 beads.
If the purple bead is in one of the four corners, then there are three adjacent cells where the green bead cannot go.
That leaves 5 cells where the green bead can go.
Once the green bead is placed, there are 7 cells left for the other 7 beads, and they can be arranged in any order.
That means there are 7! ways to place them.
So for each corner position of the purple bead, there are 5*7! ways to place the other beads.
If the purple bead is in one of the four edge cells, then there are five adjacent cells where the green bead cannot go.
That leaves 3 cells where the green bead can go.
Again, once the green bead is placed, there are 7 cells left for the other 7 beads, and they can be arranged in any order.
That means there are 7! ways to place them.
So for each edge position of the purple bead, there are 3*7! ways to place the other beads.
If the purple bead is in the center cell, then there are eight adjacent cells where the green bead cannot go.
That leaves no cells where the green bead can go.
So for the center position of the purple bead, there are no ways to place the other beads.
Therefore, the total number of ways to place the beads, given a fixed position of the purple bead, is 4*5*7! + 4*3*7! + 0 = 28*7!.
However, this counts every arrangement four times, because I can rotate the grid by 90 degrees and get the same arrangement.
To avoid this overcounting, I need to divide by 4, the number of possible rotations.
So the final answer is 28*7!/4 = 7*7! = 35280.