Question

How many three-digit numbers are not divisible by $5,$ have digits that sum to less than $20,$ and have the first digit equal to the third digit?

$52$

$60$

$66$

$68$

$70$

Note that for the number to not be divisible by $5,$ the units digits cannot be either $0$ or $5.$

Let $x$ be the hundreds and units digit and $y$ be the tens digit. Then we want $2x + y \lt 20.$ Casing on the $8$ possible values of $x,$ we get:

If $x$ is $1, 2, 3,$ or $4,$ then $y$ can be anything since $y \lt 10.$

If $x = 6,$ then $y \lt 8,$ which gives us $8$ solutions.

If $x = 7,$ then $y \lt 6,$ which gives us $6$ solutions.

If $x = 8,$ then $y \lt 4,$ which gives us $4$ solutions.

If $x = 9,$ then $y \lt 2,$ which gives us $2$ solutions.

This gives us a total of $4 \cdot 10 + 8 + 6 + 4 + 2 = 60$ solutions.

Thus, B is the correct solution.