The average of marks of fourteen students was 71. After checking it came to notice that the marks of a student were entered 42 instead of 56 and marks of another student were entered 74 instead of 32, the correct average is :

Question

Accepted Answer

To determine the correct average of the marks of the fourteen students, we need to adjust the total sum of the marks based on the corrections provided and then calculate the new average.

**Step-by-step Explanation:**

1. **Initial Calculation:**
   - The average of the marks of fourteen students was initially given as 71.
   - Therefore, the total sum of the marks for the fourteen students is:  
     $ \text{Total Sum} = 71 \times 14 = 994 $.

2. **Corrections to the Marks:**
   - One student's marks were entered as 42 instead of 56. This means the marks were underreported by $ 56 - 42 = 14 $.
   - Another student's marks were entered as 74 instead of 32. This means the marks were overreported by $ 74 - 32 = 42 $.

3. **Adjusting the Total Sum:**
   - To correct the total sum, we need to add 14 (for the underreported marks) and subtract 42 (for the overreported marks) from the initial total sum.
   - Corrected Total Sum = $ 994 + 14 - 42 = 966 $.

4. **Calculating the Correct Average:**
   - The correct average is calculated by dividing the corrected total sum by the number of students:
   - Correct Average = $ \frac{966}{14} = 69 $.

Therefore, the correct average of the marks is 69, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (71):** This was the initial average before the corrections were made. After adjusting for the errors in the marks, the average changes, so 71 is incorrect.
  
- **Option 3 (74):** This value is higher than the initial average and does not reflect the corrected total sum. Therefore, it is incorrect.
  
- **Option 4 (68):** This value is lower than the correct average calculated after adjustments. Hence, it is incorrect.

In conclusion, the correct average after making the necessary corrections is indeed 69, which is why Option 1 is the correct answer.