First slide

Combinations

Question

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

Moderate

A

15
B

16
C

17
D

18

Solution

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.
$\begin{array}{l} ^{n - 1} C_{2} = \frac{20}{100} (^{n} C_{3}) \\ \Rightarrow \frac{(n - 1) (n - 2)}{2} = \frac{1}{5} \frac{n (n - 1) (n - 2)}{(6)} \Rightarrow n = 15 . \end{array}$

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