We can solve this problem using set theory and probabilities.

Step 1: Define Events

• Let $A_1$ represent the event that a student earned an A on the final exam.
• Let $A_2$ represent the event that a student earned an A for the entire course.
• From the problem:
  • $|A_1| = 15$ (students who earned an A on the final exam)
  • $|A_2| = 10$ (students who earned an A for the entire course)
  • $|A_1 \cap A_2| = 7$ (students who earned an A on both the final exam and the entire course)

The total number of students in the class is 28.

Step 2: Use Inclusion-Exclusion to Find $|A_1 \cup A_2|$

To find the number of students who earned an A either on the final exam or for the entire course (or both), we use the inclusion-exclusion principle: $|A_1 \cup A_2| = |A_1| + |A_2| - |A_1 \cap A_2|$ Substitute the known values: $|A_1 \cup A_2| = 15 + 10 - 7 = 18$ So, 18 students earned an A on the final exam, for the course, or both.

Step 3: Find the Complement (Neither A on the Exam nor the Course)

The number of students who did not earn an A on the final exam and did not earn an A for the entire course is the complement of $A_1 \cup A_2$: [ \text{Students who earned neither} = 28 - |A_1 \cup A_2| = 28 - 18 = 10 \