For the following page reference string 4,3,2,1,4,3,5,4,3,2,1,5, the number of…

Reference string: 4, 3, 2, 1, 4, 3, 5, 4, 3, 2, 1, 5. Frame size = 3. We simulate the Least Recently Used (LRU) replacement and count page faults.

1. Step 1, reference 4: frames = [4, -, -]; page fault (total faults = 1).

2. Step 2, reference 3: frames = [4, 3, -]; page fault (total faults = 2).

3. Step 3, reference 2: frames = [4, 3, 2]; page fault (total faults = 3).

4. Step 4, reference 1: 1 is not in frames. The least recently used page is 4 (used at step 1), so replace 4 → frames = [1, 3, 2]; page fault (total faults = 4).

5. Step 5, reference 4: 4 is not in frames. LRU among [1, 3, 2] is 3 (used at step 2), replace 3 → frames = [1, 4, 2]; page fault (total faults = 5).

6. Step 6, reference 3: 3 is not in frames. LRU among [1, 4, 2] is 2 (used at step 3), replace 2 → frames = [1, 4, 3]; page fault (total faults = 6).

7. Step 7, reference 5: 5 is not in frames. LRU among [1, 4, 3] is 1 (used at step 4), replace 1 → frames = [5, 4, 3]; page fault (total faults = 7).

8. Step 8, reference 4: 4 is in frames [5, 4, 3]; hit (no page fault, total faults = 7). Update recency of 4.

9. Step 9, reference 3: 3 is in frames [5, 4, 3]; hit (no page fault, total faults = 7). Update recency of 3.

10. Step 10, reference 2: 2 is not in frames. Current recency: 5 (used step 7), 4 (step 8), 3 (step 9); LRU is 5 → replace 5 → frames = [2, 4, 3]; page fault (total faults = 8).

11. Step 11, reference 1: 1 is not in frames. Current recency: 2 (step 10), 4 (step 8), 3 (step 9); LRU is 4 (earliest) → replace 4 → frames = [2, 1, 3]; page fault (total faults = 9).

12. Step 12, reference 5: 5 is not in frames. Current recency: 2 (step 10), 1 (step 11), 3 (step 9); LRU is 3 (earliest) → replace 3 → frames = [2, 1, 5]; page fault (total faults = 10).

Total page faults: 10

Key idea: On every access update the recency; on a miss replace the page that was not used for the longest time. Following this rule for the given sequence with three frames yields 10 faults.