The correct answer is $\text{C)}$. **Step-by-step Solution:** This problem can be effectively solved using the concept of weighted averages, often visualized with the alligation rule. **Method 1: Using Weighted Average (Alligation Rule)** 1. **Identify the individual averages and the overall average:** * Average score of boys (Group 1 average) $= 70$ * Average score of girls (Group 2 average) $= 75$ * Overall average score of the class $= 72$ 2. **Apply the alligation rule:** The alligation rule states that if two groups with averages $A_1$ and $A_2$ are combined to form a larger group with an overall average $A$, then the ratio of the number of elements in the two groups, $N_1 : N_2$, is given by $|A_2 - A| : |A - A_1|$. In our case: * $A_1 = 70$ (Boys' average) * $A_2 = 75$ (Girls' average) * $A = 72$ (Overall average) The setup looks like this: latex \begin{array}{ccc} \text{Boys' Average} & & \text{Girls' Average} 70 & & 75 & \searrow & \nearrow & 72 & \swarrow & \nwarrow & (75 - 72) & : & (72 - 70) 3 & : & 2 \end{array} 3. **Determine the ratio of the number of boys to girls:** The ratio of the number of boys to the number of girls is $3:2$. Let the number of boys be $N_B$ and the number of girls be $N_G$. So, $N_B : N_G = 3:2$. 4. **Calculate the number of boys:** The total number of students in the class is $30$. The total parts in the ratio is $3 + 2 = 5$. Number of boys $N_B = \frac{\text{Ratio part for boys}}{\text{Total ratio parts}} \times \text{Total students}$ $N_B = \frac{3}{5} \times 30 = 3 \times 6 = 18$ **Method 2: Using Algebraic Equations** 1. **Define variables:** * Let $N_B$ be the number of boys. * Let $N_G$ be the number of girls. 2. **Formulate equations based on the given information:** * Total number of students: $N_B + N_G = 30$ \quad (Equation 1) * Total score of the class = Average score $\times$ Total number of students Total score of the class $= 72 \times 30 = 2160$ * The total score can also be expressed as the sum of scores of boys and girls: Total score of boys $= 70 \times N_B$ Total score of girls $= 75 \times N_G$ So, $70N_B + 75N_G = 2160$ \quad (Equation 2) 3. **Solve the system of equations:** From Equation 1, we can express $N_G$ in terms of $N_B$: $N_G = 30 - N_B$. Substitute this into Equation 2: $70N_B + 75(30 - N_B) = 2160$ $70N_B + 2250 - 75N_B = 2160$ $-5N_B = 2160 - 2250$ $-5N_B = -90$ $N_B = \frac{-90}{-5}$ $N_B = 18$ Both methods yield the same result. There are $18$ boys in the class. The final answer is $\boxed{\text{C}}$.