Final answer:
To find k in the number of distinct five-digit numbers with a 2 at the 10-thousands place, we calculate the possibilities for each place value and multiply them together, yielding 3024. Dividing this by the given 336k, we get k = 9. However, this does not match any of the answer choices.
Step-by-step explanation:
We need to find the value of k if the number of five-digit numbers with distinct digits and 2 at the 10-thousands place is 336k. Let's analyze the given problem step by step:
- Firstly, the digit at the 10-thousands place is fixed, which is 2. So, we have no choice for this place.
- For the thousands place, we have 9 options (1,3,4,5,6,7,8,9).
- For the hundreds place, we now have 8 options, because we cannot repeat any of the two digits already used.
- For the tens place, this drops to 7 options.
- And finally, for the units place, we're left with 6 options.
To calculate the total number of five-digit numbers possible, we multiply these choices together:
9 x 8 x 7 x 6 = 3024
The problem states that the total number of such numbers is 336k. We equate this to our calculated total:
3024 = 336k
Dividing both sides by 336, we find the value of k:
k = 3024 / 336 = 9
Therefore, our initial equation seems to be incorrect because none of the choices (1, 2, 3, 4) match this result. There may be a mistake in the original problem or in our interpretation. Thus, it's best to double-check the parameters or the arithmetic involved.