Question

A class consist of 10 boys and 8girls. Three students are selected at random what is the probability that the selected group has 1 boy and 2girls?
A $\dfrac{{34}}{{103}}$
B $\dfrac{{35}}{{102}}$
C $\dfrac{{36}}{{107}}$
D None of these

Answer 1

Hint: In this problem, first we need to obtain the combination of selecting three students among 18 students. Next, find the combination of selecting 1boys and 2 girls among 18 students and hence find the probability.

Complete step-by-step solution -
The total no of students in the class is 18.
Selection of three students among 18 students is calculated as follows:
${}^{18}{C_3} \\$
$\Rightarrow \,\dfrac{{18!}}{{3! \times \left( {18 - 3} \right)!}} \\$
$\Rightarrow \,\dfrac{{18 \times 17 \times 16 \times 15!}}{{3! \times 15!}} \\$
$\Rightarrow \dfrac{{\,18 \times 17 \times 16}}{{3 \times 2}} \\$
$\Rightarrow \,816 \\ $
Selection of 1boys and 2 girls among 18 students is shown below:
  ${}^{10}{C_1} \times {}^8{C_2} \\$
 $ \Rightarrow \,\dfrac{{10!}}{{1! \times \left( {10 - 1} \right)!}} \times \dfrac{{8!}}{{2! \times \left( {8 - 2} \right)!}} \\$
  $ 4\Rightarrow \dfrac{{10!}}{{9!}} \times \dfrac{{8!}}{{2! \times 6!}} \\$
  $ \Rightarrow \,\dfrac{{10 \times 9!}}{{9!}} \times \dfrac{{8 \times 7 \times 6!}}{{2! \times 6!}} \\$
 $ \Rightarrow \,\dfrac{{10 \times 8 \times 7}}{2} \\$
 $ \Rightarrow \,280 \\$
The probability of selection of group of one boys and two girls is calculated as follows:
$\dfrac{{{}^{10}{C_1} \times {}^8{C_2}}}{{{}^{18}{C_3}}} \\ $
$ \Rightarrow \dfrac{{280}}{{816}} \\$
$ \Rightarrow \dfrac{{35}}{{102}} \\ $
Thus, the probability of selection of a group of one boy and two girls is $\dfrac{{35}}{{102}}$.

Note: The probability of selection of a group of one boy and two girls are obtained by dividing the combination of 3 students among 18 students by a combination of 1 boy and 2 girls among 18 students.