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If 20 \% of three subsets (i.e., subsets containing exactly three elements) of the set \mathrm{A=\left\{a_{1}, a_{2}, \ldots\right.a_{n} ) } contain \mathrm{a_{1} },  then value of \mathrm{n} is

Option: 1

15


Option: 2

16


Option: 3

17


Option: 4

18


Answers (1)

best_answer

The number of subsets of \mathrm{A} containing exactly three elements is \mathrm{{ }^{n} C_{3}} whereas the number of three subsets of A that contain \mathrm{a_{1}}, is \mathrm{{ }^{n-1} C_{2}}. We are given,

\begin{aligned} { }^{n-1} C_{2} & =\frac{20}{100}\left({ }^{n} C_{3}\right) \\ \Rightarrow \quad \frac{(n-1)(n-2)}{2} & =\frac{1}{5} \frac{n(n-1)(n-2)}{(6)} \\ \Rightarrow \quad n=15 . & \end{aligned}

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