Data Sufficiency

For a shorter explanation:

There are x high-level officials (where x is a positive integer). Each high-level official supervises x^2 mid-level officials, each of whom, in turn, supervises x^3 low-level officials. How many high-level officials are there?
(1) There are fewer than 60 low-level officials.
(2) No official is supervised by more than one person.

Solution:

Assuming that no official is supervised by more than one person:

# of HL officials = x (from the given)
# of ML officials = x(x^2) = x^3 (by the fundamental counting principle or, by conversion (x^2 ML per HL)(no. of HL) = total ML )
# of LL officials: = x^3(x^3) = x^6 (by the fundamental counting principle or, by conversion (x^3 LL per ML)(no. of ML) = total HL )


(1) We do not know whether one official can supervise more than 1 person. If at least 1 official can supervise more than 1 person, our equations cannot hold because the total number of HL, ML, or LL will be lower and we cannot determine this by with the current information given.

(2) The statement is useless by itself.

(1) and (2). Since no official is supervised by more than one person, we can use our equations, particularly, the one for the # of LL:
# of LL = x^6 < 60

Since x is an integer, the only x that can make the equation true is when x = 1.

Therefore, the answer is C.

Let's assume that there are 2 high level officials. This means that each of these 2 high level officials supervises 4 (or x^2) mid-level officials, and that each of these 4 mid-level officials supervises 8 (or x^3) low-level officials.

It is possible that the supervisors do not share any subordinates. If this is the case, then, given 2 high level officials, there must be 2(4) = 8 mid-level officials, and 8(8) = 64 low-level officials.

Alternatively, it is possible that the supervisors share all or some subordinates. In other words, given 2 high level officials, it is possible that there are as few as 4 mid-level officials (as each of the 2 high-level officials supervise the same 4 mid-level officials) and as few as 8 low-level officials (as each of the 4 mid-level officials supervise the same 8 low-level officials).

Statement (1) tells us that there are fewer than 60 low-level officials. This alone does not allow us to determine how many high-level officials there are. For example, there might be 2 high level officials, who each supervise the same 4 mid-level officials, who, in turn, each supervise the same 8 low-level officials. Alternatively, there might be 3 high-level officials, who each supervise the same 9 mid-level officials, who, in turn, each supervise the same 27 low-level officials.

Statement (2) tells us that no official is supervised by more than one person, which means that supervisors do not share any subordinates. Alone, this does not tell us anything about the number of high-level officials.

Combining statements 1 and 2, we can test out different possibilities.

If x = 1, there is 1 high-level official, who supervises 1 mid-level official (12 = 1), who, in turn, supervises 1 low-level official (13 = 1).

If x = 2, there are 2 high-level officials, who each supervise a unique group of 4 mid-level officials, yielding 8 mid-level officials in total. Each of these 8 mid-level officials supervise a unique group of 8 low-level officials, yielding 64 low-level officials in total. However, this cannot be the case since we are told that there are fewer than 60 low-level officials.

Therefore, based on both statements taken together, there must be only 1 high-level official. The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.