The administration has increased the number of fines for mining safety violations as part of their campaign to protect miners.
Key to getting this one right is being aware that the quantity word number is used when the quantity is made up of countable items, such as fines, baseballs, or apples, whereas amount is used the the quantity is made up of something not really countable such as water or apple juice. So you would say number of baseballs and amount of water.
a) has increased the number of fines for mining safety violations as part of their
Here number is correctly used with the countable fines, but their at the end does not match the singular antecedent, administration.
b) have increased the number of fines for mining safety violations as part of their
Ok the game goes on, with this one out because of have and their not working with administration.
c) has increased the number of fines for mining safety violations as part of its
This is the last to use number. So it had better be right, and it is. Has and its work with administration.
d) has increased the amount of fines for mining safety violations as part of its
Amount does not work with the countable fines.
e) have increased the amount of fines for mining safety violations as part of their
Amount does not work with the countable fines.
So the correct answer is C.
I want to add a caveat. D and E were pretty easy to eliminate on the basis of the incorrect use of amount. Just be careful when using splits like that one though, because multiple things in a SC answer choice can be changed, so that what seems like a split may not be. For instance, in this question, seeing the word amount is not quite sufficient for elimination. You would want to be sure that fines is still present so that amount does not work.