To determine how many times the second, minute, and hour hands of the clock coincide in a 12-hour period from 3 PM to 3 AM, we need to analyze the behavior of the clock hands. When we say that the hands "coincide," we mean that all three hands are exactly on top of each other.

On a standard 12-hour clock, the hour, minute, and second hands coincide 11 times in a 12-hour duration. Here is how this happens:

1. All three hands coincide at exactly 12 o'clock.

The hands coincide again approximately every 65.4545 minutes.

To understand why the hands coincide 11 times in 12 hours, consider the following points:

1. Initial coincidence: At the start time (e.g., 3 PM), none of the hands are coinciding.

Regular intervals: They coincide at specified intervals, as mentioned earlier.

If we start observing from 12 o'clock, the coincidences occur approximately at:

No.	Time
1	12:00
2	1:05
3	2:11
4	3:16
5	4:22
6	5:27
7	6:33
8	7:38
9	8:44
10	9:49
11	10:55

At each time listed above, the three hands coincide.

Hence, from 3 PM on one day to 3 AM the next day, the total number of times the three hands coincide is 11.

Therefore, the answer is Option A: 11.