Question

In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is:
a) $\dfrac{21}{46}$
b) $\dfrac{25}{117}$
c) $\dfrac{1}{50}$
d) $\dfrac{3}{25}$

Answer 1

Hint: We will first find the number of ways of selecting 3 out of 25, then we will find the number of ways of selecting 1 girl out of 10 girl and 2 boys out of 15, then we will find probability required as $\dfrac{\text{Number of ways of selecting 2 boys and 1 girl}}{\text{Total Sample}}$, also we will use the formula ${}^{n}{{c}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$.

Complete step-by-step answer:
It is given in the question that there are 15 boys and 10 girls, three students are selected at random, then we have to find the probability that 1 girl and 2 boys are selected at random.
Let us assume a sample space (S), where the number of ways of selecting 3 students out of 25 is $n\left( s \right)={}^{25}{{C}_{3}}$. Using the formula ${}^{n}{{c}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$, we get
${}^{25}{{C}_{3}}=\dfrac{25!}{3!\left( 25-3 \right)!}=\dfrac{25\times 24\times 23\times 22!}{3!\times 22!}$, solving further, we get
\[{}^{25}{{C}_{3}}=\dfrac{25\times 24\times 23}{3\times 2\times 1}=\dfrac{25\times 24\times 23}{6}=25\times 4\times 23=2300\].
Now, the number of ways of selecting 1 girl out of 10 girls will be $={}^{10}{{C}_{1}}=10$ ( using the formula ${}^{n}{{c}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$).
Similarly, the number of ways of selecting 2 boys out of 15 will be $={}^{15}{{C}_{2}}=\dfrac{15!}{2!\left( 15-2 \right)!}=\dfrac{15\times 14}{2}=105$. Thus, the required probability that 1 girl and 2 boys are selected at random is given by $P=\dfrac{\left( \text{Number of ways of selecting 1 girl} \right)\times \left( \text{Number of ways of selecting 2 boys} \right)}{\text{Number of ways of selecting 3 students out of 25}}$, on putting the values, we get
$P=\dfrac{10\times 105}{2300}=\dfrac{1050}{2300}=\dfrac{105}{230}=\dfrac{21}{46}$.
Therefore, option a) is the correct answer.

Note: Students may find a number of ways of selecting boys among all 25 students and similarly, a number of ways of selecting girls among all 25 students which is not correct at all. We have to find the number of ways of selecting boys among 15 and the number of ways of selecting girls among 10 and not among 25.