How many times the hour hand and the minute hand coincide in a clock between 10:00 a.m. and 2:00 p.m. (same day)?

The correct option is 3 times.

To understand why this is correct, let us analyze the relative movement of the hour hand and the minute hand of a clock.

In a standard 12-hour clock, the hour hand and the minute hand coincide exactly 11 times in every 12-hour cycle (not 12 times). This is because the hands do not overlap exactly once every 60 minutes, but rather once every:
$$\frac{60}{\frac{11}{12}}=\frac{720}{11}\approx 65.45\text{minutes}$$
(or approximately 65 minutes and 27 seconds).

Let us look at the specific intervals between 10:00 a.m. and 2:00 p.m. on the same day:
1. Between 10:00 a.m. and 11:00 a.m.: The hands coincide once, at approximately 10:54 a.m.
2. Between 11:00 a.m. and 1:00 p.m.: The hands coincide exactly once in this 2-hour span, which occurs precisely at 12:00 p.m. (noon). Note that they do not overlap at any other time between 11:00 a.m. and 12:00 p.m., nor between 12:00 p.m. and 1:00 p.m.
3. Between 1:00 p.m. and 2:00 p.m.: The hands coincide once, at approximately 1:05 p.m.

Adding these occurrences together:
1 (between 10:00 a.m. and 11:00 a.m.) + 1 (at 12:00 p.m.) + 1 (between 1:00 p.m. and 2:00 p.m.) = 3 times.

Therefore, the hour hand and the minute hand coincide exactly 3 times between 10:00 a.m. and 2:00 p.m.