Using the Fundamental Counting Theorem, it is found that there are 648 ways to paint the spinner.
It is a theorem that states that if there are n things, each with [tex]n_1, n_2, \cdots, n_n[/tex] ways to be done, each thing independent of the other, the number of ways they can be done is:
[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]
In this problem, we have that the first sector can be painted in any of the 4 colors, the others until the 5th can be painted in 3 colors(not the adjacent), and the sixth in only 2, as it is adjacent to both the 5th and the 1st sectors, hence:
[tex]n_1 = 4, n_2 = n_3 = n_4 = n_5 = 3, n_6 = 2[/tex]
Hence the number of ways is given by:
N = 4 x 3 x 3 x 3 x 3 x 2 = 648.
More can be learned about the Fundamental Counting Theorem at https://brainly.com/question/24314866
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