If $20 %$ of all the subsets with exactly 3 elements(i.e. subsets containing exactly three elements) of the set $A = (a_{1}, a_{2}, . . . . . . . a_{n})$ contain $a_{1}$ , then value of $n$ is

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Solution

The correct option is A 15
The number of subsets of $A$ containing exactly three elements is $^{n} C_{3}$
whereas the number of three subsets of $A$ that contain $a_{1}$ is $^{n - 1} C_{2}$
We are given, $^{n - 1} C_{2} = \frac{20}{100} (^{n} C_{3})$
$\Rightarrow \frac{(n - 1) (n - 2)}{2} = \frac{1}{15} \frac{n (n - 1) (n - 2)}{6} \Rightarrow n = 15$

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