How many such 5's are there in the following number sequence, each of which is immediately preceded by 3 or 4 but not immediately followed by 8 or 9?

3 5 9 5 4 5 5 3 5 8 4 5 6 7 3 5 7 5 5 4 5 2 3 5 1 0

The sequence given is:

$3, 5, 9, 5, 4, 5, 5, 3, 5, 8, 4, 5, 6, 7, 3, 5, 7, 5, 5, 4, 5, 2, 3, 5, 1, 0$

The criteria for counting a $5$:

• Must be immediately preceded by $3$ or $4$
• Must NOT be immediately followed by $8$ or $9$

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Let's number each position and examine every $5$ in the sequence:

Position $1$ to $26$: $3, 5, 9, 5, 4, 5, 5, 3, 5, 8, 4, 5, 6, 7, 3, 5, 7, 5, 5, 4, 5, 2, 3, 5, 1, 0$

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Position $2$: $3 \rightarrow \boxed{5} \rightarrow 9$

Preceded by $3$ ✓, but followed by $9$ ✗

Does not count

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Position $4$: $9 \rightarrow \boxed{5} \rightarrow 4$

Preceded by $9$ ✗

Does not count

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Position $6$: $4 \rightarrow \boxed{5} \rightarrow 5$

Preceded by $4$ ✓, followed by $5$ ✓

Counts ✓

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Position $7$: $5 \rightarrow \boxed{5} \rightarrow 3$

Preceded by $5$ ✗

Does not count

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Position $9$: $3 \rightarrow \boxed{5} \rightarrow 8$

Preceded by $3$ ✓, but followed by $8$ ✗

Does not count

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Position $12$: $4 \rightarrow \boxed{5} \rightarrow 6$

Preceded by $4$ ✓, followed by $6$ ✓

Counts ✓

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Position $16$: $3 \rightarrow \boxed{5} \rightarrow 7$

Preceded by $3$ ✓, followed by $7$ ✓

Counts ✓

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Position $18$: $7 \rightarrow \boxed{5} \rightarrow 5$

Preceded by $7$ ✗

Does not count

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Position $19$: $5 \rightarrow \boxed{5} \rightarrow 4$

Preceded by $5$ ✗

Does not count

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Position $21$: $4 \rightarrow \boxed{5} \rightarrow 2$

Preceded by $4$ ✓, followed by $2$ ✓

Counts ✓

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Position $24$: $3 \rightarrow \boxed{5} \rightarrow 1$

Preceded by $3$ ✓, followed by $1$ ✓

Counts ✓

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The $5$'s that satisfy both conditions are at positions $6, 12, 16, 21,$ and $24$.

Therefore, there are $5$ such $5$'s in the sequence.