Answer: B — 19

Count the digit $5$ separately in tens and units places of two-digit numbers $10$–$99$.

Tens place. The tens digit is $5$ in $50, 51, \ldots, 59$ — that's $10$ occurrences.

Units place. The units digit is $5$ in $15, 25, 35, 45, 55, 65, 75, 85, 95$ — that's $9$ occurrences.

Total: $10 + 9 = 19$.