The appropriate method to use in solving the given question is permutation. Then, the numbers of ways to arrange the buttons is 1320 ways.
In the given question, the order of arrangement is important, so that the number of ways that Fiona could order the buttons can be determined by permutation method.
i.e [tex]_{n}[/tex][tex]P_{r}[/tex] = [tex]\frac{n!}{(n - r)!}[/tex]
Red 6
Green 3
Yellow 3
Total 12
Since no two neighbouring buttons of the same colour is allowed, then 3 buttons would be arranged at a time.
Thus, n = 12 and r = 3;
[tex]_{12}[/tex][tex]P_{3}[/tex] = [tex]\frac{12!}{(12 - 3)!}[/tex]
= [tex]\frac{12!}{9!}[/tex]
= [tex]\frac{12*11*10*9!}{9!}[/tex]
= 12 x 11 x 10
[tex]_{12}[/tex][tex]P_{3}[/tex] = 1320
Therefore the buttons can be arranged in 1320 ways without two neighbors having the same colour.
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Answer:
Total number of ways are [tex]\[40\][/tex]
Step-by-step explanation:
[tex]\[6\][/tex] Red balls can only be placed at alternate spots.
This can be done in two different ways, by starting at spot one and by starting at spot two.
Now, [tex]\[3\][/tex] yellow [tex]\[3\][/tex] green balls can be placed in the intermediate spots.
It can be done in following ways:
[tex]\[=\frac{6!}{3!3!}=20\][/tex]
So, total number of ways are= [tex]\[2\times 20=40\][/tex]
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