The correct answer is B. Here's a step-by-step solution: Step 1: Define variables and initial conditions. Let the original number of students in the class be $N$. The original average score of these $N$ students was $60$. Step 2: Calculate the total score of the original students. We know that $\text{Average} = \frac{\text{Sum}}{\text{Count}}$, so $\text{Sum} = \text{Average} \times \text{Count}$. $\text{Total Score}_{\text{original}} = N \times 60 = 60N$ Step 3: Calculate the total score of the new students. Number of new students = $10$ Average score of new students = $75$ $\text{Total Score of New Students} = 10 \times 75 = 750$ Step 4: Determine the total number of students after the new students joined. $\text{Total Number of Students}_{\text{new}} = \text{Original Students} + \text{New Students}$ $\text{Total Number of Students}_{\text{new}} = N + 10$ Step 5: Calculate the new total score of all students. After the new students joined, the overall average score became $65$. $\text{Total Score}_{\text{new}} = \text{Total Number of Students}_{\text{new}} \times \text{Overall Average Score}$ $\text{Total Score}_{\text{new}} = (N + 10) \times 65 = 65(N + 10)$ Step 6: Formulate and solve the equation. The total score of the new group is the sum of the total scores of the original students and the new students. $\text{Total Score}_{\text{original}} + \text{Total Score of New Students} = \text{Total Score}_{\text{new}}$ $60N + 750 = 65(N + 10)$ Expand the right side of the equation: $60N + 750 = 65N + 650$ Rearrange the terms to solve for $N$ by collecting $N$ terms on one side and constant terms on the other: $750 - 650 = 65N - 60N$ $100 = 5N$ Divide both sides by $5$: $N = \frac{100}{5}$ $N = 20$ Step 7: Conclusion. The original number of students in the class was $20$. Alternatively, using the concept of a weighted average: Let $N_1$ be the original number of students and $A_1$ be their average. So, $N_1 = N$ and $A_1 = 60$. Let $N_2$ be the number of new students and $A_2$ be their average. So, $N_2 = 10$ and $A_2 = 75$. Let $A_{\text{final}}$ be the final average of all students, which is $65$. The formula for weighted average is: $A_{\text{final}} = \frac{N_1 A_1 + N_2 A_2}{N_1 + N_2}$ Substitute the given values into the formula: $65 = \frac{N \times 60 + 10 \times 75}{N + 10}$ $65 = \frac{60N + 750}{N + 10}$ Multiply both sides by $(N + 10)$: $65(N + 10) = 60N + 750$ $65N + 650 = 60N + 750$ Subtract $60N$ from both sides: $5N + 650 = 750$ Subtract $650$ from both sides: $5N = 750 - 650$ $5N = 100$ Divide by $5$: $N = 20$ Both methods confirm that the original number of students was $20$. Therefore, Option B is the correct choice.