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# From a class of 12 girls and 18 boys, two students are chosen randomly. What is the probability that both of them are girls?${\text{A}}{\text{. }}\dfrac{{22}}{{145}} \\ {\text{B}}{\text{. }}\dfrac{{13}}{{15}} \\ {\text{C}}{\text{. }}\dfrac{1}{{18}} \\ {\text{D}}{\text{. }}\dfrac{1}{{15}} \\$

Last updated date: 20th Jun 2024
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Hint- In this question we have to find the probability that when two students are chosen randomly both of them are girls. So, to solve this we will use combinations and formulas for probability to reach the answer.

So, $n\left( A \right) = {}^{12}{C_2}$ as we have to select 2 girls out of 12 girls of the class.
So, $n\left( S \right) = {}^{30}{C_2}$ as we have to select 2 students out of 30 students in the class.
${\text{Probability}} = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$
${\text{Probability}} = \dfrac{{{}^{12}{C_2}}}{{{}^{30}{C_2}}} = \dfrac{{12!}}{{2! \times \left( {12 - 2} \right)!}} \times \dfrac{{2! \times \left( {30 - 2} \right)!}}{{30!}}$
$\Rightarrow {\text{Probability}} = \dfrac{{12 \times 11}}{{30 \times 29}} = \dfrac{{22}}{{145}}$
Hence, the correct answer is A. $\dfrac{{22}}{{145}}$