Official Solution: Write \(x\) for the number of hats Xander has, \(y\) for the number of hats Yolanda has, and \(z\) for the number of hats Zelda has. From the question stem, we know that \(x \lt y \lt z\) and that \(x + y + z = 12\). Moreover, since each person has at least one hat, and people can only have integer numbers of hats, we know that \(x\), \(y\), and \(z\) are all positive integers. With this number of constraints, we should go ahead and list scenarios that fit all the constraints. Start with \(x\) and \(y\) as low as possible, then adjust from there, keeping the order, keeping the sum at 12, and ensuring that no two integers are the same.

Scenario \(x\) \(y\) \(z\) (a) 1 2 9 (b) 1 3 8 (c) 1 4 7 (d) 1 5 6 (e) 2 3 7 (f) 2 4 6 (g) 3 4 5

These are the only seven scenarios that work. As a reminder, we are looking for the value of \(y\). Now, we turn to the statements.

Statement (1): INSUFFICIENT. We are told that \(z - x\) is less than or equal to 5. This rules out scenarios (a) through (c), but the last four scenarios still work. Thus, \(y\) could be 3, 4, or 5.

Statement (2): INSUFFICIENT. We are told that \(xyz\) is less than 36. We work out this product for the seven scenarios:

(a) 18

(b) 24

(c) 28

(d) 30

(e) 42

(f) 48

(g) 60

We can rule out scenarios (e) through (g), but (a) through (d) still work. Thus, \(y\) could be 2, 3, 4, or 5.

Statements (1) and (2) together: SUFFICIENT. Only scenario (d) survives the constraints of the two statements. Thus, we know that \(y\) is 5.

Answer: C

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