Question

There are seven boys and three girls on a school tennis team. The coach must select four people from this group to participate in county championship. What is the probability that two boys and two girls are chosen for the team?

• A. $\frac{3}{10}$
• B. $\frac{7}{10}$
• C. $\frac{2}{3}$
• D. None of the above

Answer 1

The correct option is A $\frac{3}{10}$

Given:
Total number of boys $=7$
Total number of girls $=3$

Step 1: $2$ boys can be selected by ${}^7{\text{C}}_2$ ways
$\Rightarrow {}^7{\text{C}}_2=21$

Step 2: $2$ girls can be selected by ${}^3{\text{C}}_2$ ways
$\Rightarrow {}^3{\text{C}}_2=3$

Step 3: $4$ member can be selected for the team by ${}^{10}{\text{C}}_4$ ways
$\Rightarrow {}^{10}{\text{C}}_4=\frac{10\times 9\times 8\times 7}{4\times 3\times 2}$
$=210$

Step 4: Probability of selecting $4$ member for the team
$=\frac{{}^7{\text{C}}_2\times {}^3{\text{C}}_2}{{}^{10}{\text{C}}_4}$
$=\frac{21\times 3}{210}$
$=\frac{3}{10}$