Question

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

Answer 1

20,160

Explanation:

The arrangement of the 9 differently colored bead can be presented as follows;

`\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right]`

Where 1 is the purple bead and 2 is the green bead, the number of ways of arrangement where the green bead cannot be adjacent to the green either horizontally, vertically, or diagonally

Placing the purple bead at 1, the location of the green bead = 3, 6, 7, 8, or 9

The number of ways = 5 ways × 7! ways of arranging the other beads

With the purple bead at 2, the location of the green bead = 7, 8, or 9

The number of ways = 3 × 7!

With the purple at 3, we also have 5 × 7! ways

At 4, similar to 2, we have, 3 × 7! ways

At 5, we have, 0 × 7!

At 6, we have 3 × 7!

For 7, 8, and 9, we have, (5 + 3 + 5) × 7!

The total number of ways = (5 + 3 + 5 + 3 + 0 + 3 + 5 + 3 + 5) × 7! ways

However, placing the purple bead at 1, 2, 3, 4, 6, 7, 8, and 9, (8 positions) can be taken as reflection and rotation of each other and can be considered the same

Therefore, the total number of acceptable ways = (5 + 3 + 5 + 3 + 0 + 3 + 5 + 3 + 5) × 7!/8 = 20,160 ways