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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

Ria has 4 green marbles and 8 red marbles. She arranges them in a...

Ria has 4 green marbles and 8 red marbles. She arranges them in a circle randomly. If the probability that no two green marbles are adjacent is pq\frac{p}{q} where the positive integers p,qp, q have no common factors other than 1 , what is p+qp+q ?

Correct answer

40

Step-by-step solution

Favourable ways =7!8C4⋅4=7!^8 \mathrm{C}_4 \cdot 4 ! Sample space =11=11 ! Pr⁡=7!8!4!4!×4!11!=7⋅6⋅5⋅4!4!⋅11⋅10⋅9=733∴p+q=40\begin{aligned} & \operatorname{Pr}=\frac{7!8!}{4!4!} \times \frac{4!}{11!}=\frac{7 \cdot 6 \cdot 5 \cdot 4!}{4!\cdot 11 \cdot 10 \cdot 9}=\frac{7}{33} \\ & \therefore p+q=40 \end{aligned}
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