1. Interpret the question

A five-digit number has places:

• ten-thousands
• thousands
• hundreds
• tens
• units

The statement says digit 2 is at the 10th place, i.e. 2 is in the tens place.

So the number looks like:

$\overline{abcd e}$

with

$d = 2.$

Also, all digits are distinct.

2. Fix the tens digit

Since the tens digit is already fixed as $2$, we now choose the remaining four digits for:

• ten-thousands place
• thousands place
• hundreds place
• units place

from the remaining digits $\{0,1,3,4,5,6,7,8,9\}$.

There are $9$ digits available.

3. Choose the first digit carefully

The first digit of a five-digit number cannot be $0$.

So for the ten-thousands place, possible choices are:

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

Thus number of choices for the first digit is:

$8$

4. Fill the remaining places

After choosing the first digit, we have already used:

• digit $2$
• one more chosen digit

So remaining available digits for the thousands, hundreds, and units places are:

$8, 7, 6$

respectively.

Hence total count is:

$8 \times 8 \times 7 \times 6$

Calculate:

$8 \times 8 \times 7 \times 6 = 2688$

5. Compare with given form

Given number of such numbers is:

$336k$

So,

$336k = 2688$

Therefore,

$k = \frac{2688}{336} = 8$

6. Final answer

$\boxed{8}$

So the correct option is B.