For the following page reference string $4,3,2,1,4,3,5,4,3,2,1,5$, the number of page faults that occur in the least Recently Used $(LRU)$ page replacement algorithm with frame size $3$ is:

Given page reference string is:

$4,3,2,1,4,3,5,4,3,2,1,5$

Frame size $=3$

Using LRU page replacement, the least recently used page is replaced when a page fault occurs.

Now process each page:

$4$ gives page fault.

Frames: $4$

Total faults $=1$

$3$ gives page fault.

Frames: $4,3$

Total faults $=2$

$2$ gives page fault.

Frames: $4,3,2$

Total faults $=3$

$1$ gives page fault. Replace least recently used page $4$.

Frames: $1,3,2$

Total faults $=4$

$4$ gives page fault. Replace least recently used page $3$.

Frames: $1,4,2$

Total faults $=5$

$3$ gives page fault. Replace least recently used page $2$.

Frames: $1,4,3$

Total faults $=6$

$5$ gives page fault. Replace least recently used page $1$.

Frames: $5,4,3$

Total faults $=7$

$4$ is already present.

Frames: $5,4,3$

Total faults $=7$

$3$ is already present.

Frames: $5,4,3$

Total faults $=7$

$2$ gives page fault. Replace least recently used page $5$.

Frames: $2,4,3$

Total faults $=8$

$1$ gives page fault. Replace least recently used page $4$.

Frames: $2,1,3$

Total faults $=9$

$5$ gives page fault. Replace least recently used page $3$.

Frames: $2,1,5$

Total faults $=10$

Therefore, number of page faults $=10$.