QUESTION

# In a class there are 10 boys and 8 girls. When 3 students are selected at random, the probability that 2 girls and 1 boy are selected is.A. $\dfrac{{35}}{{102}}$ B. $\dfrac{{15}}{{102}}$ C. $\dfrac{{55}}{{102}}$ D. $\dfrac{{25}}{{102}}$

Hint: Here we will use the concept of combinations and probability to select 2 girls and 1 boy from a group of 10 boys and 8 girls. The selection of n people from a group of r people is given by $^n{C_r}$ . If the selection is from different groups , just multiply their number of ways of individual selection.Then probability is given by ratio of number of ways of selecting 2 girls and 1 boy to the number of ways of selecting 3 students out of 18, we get the desired answer.

Given:
Number of boys=10
And number of girls=8
Hence, total number of students=10+8=18
Now, the number of ways selecting 2 girls out of 8=$^8{C_2}$
Now, the number of ways selecting 1 boy out of 10=$^{10}{C_1}$
Now, the number of ways selecting 2 girls and 1 boy=$^3{C_2}{ \times ^{10}}{C_1}$
Similarly, Number of ways selecting 3 students out of 18=$^{18}{C_3}$
$\Rightarrow$Probability$\left( p \right)$ =$\dfrac{{{\text{Favourable ways}}}}{{Total{\text{ number of ways}}}}$
P=$\dfrac{{Number{\text{ ways selecting 2 girls and 1 boy}}}}{{Numbers{\text{ of ways selecting 3 students out of 18}}}} = \dfrac{{^8{C_2}{ \times ^{10}}{C_1}}}{{^{18}{C_3}}}$
Now we know $^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)! \times r!}}$
$\ \Rightarrow p = \dfrac{{^8{C_2}{ \times ^{10}}{C_1}}}{{^{18}{C_3}}} = \dfrac{{\dfrac{{8!}}{{6! \times 2!}} \times \dfrac{{10!}}{{9! \times 1!}}}}{{\dfrac{{18!}}{{15! \times 3!}}}} = \dfrac{{8! \times 10! \times 15! \times 3!}}{{6! \times 2! \times 9! \times 1! \times 18!}} = \dfrac{{8 \times 7 \times 6! \times 10 \times 9! \times 15! \times 3 \times 2!}}{{6! \times 2! \times 9! \times 18 \times 17 \times 16 \times 15!}} \\ \Rightarrow p = \dfrac{{8.7.10.3}}{{18.17.16}} = \dfrac{{35}}{{102}} \\$
Therefore option A is correct.

Note: In this type of question first calculate the favourable ways, then calculate the total number of ways, then divide them we will get the required probability.Students make mistakes while calculating the probability i.e total number of ways of selecting 3 students from 18 i.e (10+8).So,have to take care while solving such type of questions.Students should remember the formula of combination, definition of probability and have to take care while selecting number of people from group by satisfying given condition.