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Explanation:

There must be a blue and a green bead between two red beads, we cannot choose one the two colors as red, and

There is at least one Green bead between any two Blue beads along the same row or column.

∴ We can have maximum two red beads in any row or column.

Consider the following configuration

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∴ The maximum number of red beads that can be used is 9.

Hence, 9.