ProbabilityHardBloom L3
Question
From a group of ' $n$ ' persons arranged in a circle 3 persons are selected at random. If the probability that no two adjacent persons are selected is $\frac{2}{7}$, then $n =$
Options
A.7
B.8
C.9
D.10
Solution
Sol. [Key Concept: $n$ persons in a circle, choose 3: probability none are adjacent.]
$$\begin{matrix} & \frac{2}{7} = \frac{\ ^{n}C_{1}^{n}\ ^{4}C_{2}}{3^{n}C_{3}} \\ & \frac{2}{7} = \frac{n(n - 4)(n - 5)}{n(n - 1)(n - 2)} = \frac{(n - 4)(n - 5)}{(n - 1)(n - 2)} \\ \Rightarrow & 5n^{2} - 57n + 136 = 0 \\ \Rightarrow \ & n = 8 \end{matrix}$$
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