The minute-hand and second-hand of a clock cross each other _______ times between 09:15:00 AM and 09:45:00 AM on a day.

The correct option is 30.

Let us analyze the motion of the second-hand and the minute-hand of a clock between 09:15:00 AM and 09:45:00 AM.

The time interval is exactly 30 minutes, starting at 09:15:00 AM and ending at 09:45:00 AM.

During each complete minute, the second-hand makes one full rotation (360 degrees) around the clock face, starting from the 12 o'clock position (0 seconds) and returning to it (60 seconds). Meanwhile, the minute-hand moves extremely slowly, covering only 0.5 degrees per minute. Because the second-hand sweeps through the entire 360 degrees while the minute-hand is nearly stationary, the second-hand must cross the minute-hand exactly once during every 60-second cycle.

Let us examine the boundary conditions at the start and end of the interval:
At exactly 09:15:00 AM, the minute-hand is at the 15-minute mark (which corresponds to the 3 position on the clock dial). However, because 15 minutes have passed since 9:00 AM, the minute-hand has actually moved slightly past the 15-minute mark (specifically, 15/60 of the way to the next minute, which is 15.25 minutes). The second-hand, at 09:15:00 AM, is exactly at the 12 o'clock position (0 seconds). Therefore, at the exact start time of 09:15:00 AM, the hands are not crossing.
Similarly, at exactly 09:45:00 AM, the second-hand is at the 12 o'clock position, while the minute-hand is slightly past the 45-minute mark. Thus, they do not cross at the exact end time either.

Since the duration is 30 minutes, the second-hand completes 30 full rotations. In each of these 30 distinct 1-minute intervals, the second-hand crosses the slowly moving minute-hand exactly once.
Specifically, the crossings occur during the following intervals:
1. Between 09:15:00 AM and 09:16:00 AM (crossing occurs shortly after the 15-second mark)
2. Between 09:16:00 AM and 09:17:00 AM (crossing occurs shortly after the 16-second mark)
... and so on, up to:
30. Between 09:44:00 AM and 09:45:00 AM (crossing occurs shortly after the 44-second mark)

Counting all these intervals, we find that the second-hand and the minute-hand cross each other exactly:

$30$