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 (ei

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

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RobertKer 3 years 2021-08-01T20:03:32+00:00 1 Answers 5 views 0

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  1. Nho
    0
    2021-08-01T20:04:57+00:00
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    Answer:

    20,160

    Step-by-step 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

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