Correct Option (C)

To determine the time when the minute hand is 3 minutes ahead of the hour hand between 6 p.m. and 7 p.m., we calculate the relative positions and speeds of the clock hands.

The minute hand moves at 6 degrees per minute.
The hour hand moves at 0.5 degrees per minute.
The relative speed of the minute hand with respect to the hour hand is 5.5 degrees per minute (6 - 0.5).

At 6:00 p.m., the hour hand is at the 6-mark (180 degrees from the 12-mark), and the minute hand is at the 12-mark (0 degrees). The hour hand is initially 180 degrees ahead of the minute hand.

For the minute hand to be 3 minutes ahead of the hour hand, it needs to cover the initial 180-degree gap and then move an additional 3 minutes (which is 3 × 6 = 18 degrees) ahead of the hour hand's current position.

Let 't' be the number of minutes past 6:00 p.m.

Angle of the minute hand from 12: 6t degrees.
Angle of the hour hand from 12: 180 degrees (initial position at 6) + 0.5t degrees (movement in 't' minutes).

We set up the equation where the minute hand's angle is 18 degrees greater than the hour hand's angle:

6t = (180 + 0.5t) + 18

6t = 198 + 0.5t

5.5t = 198

t = 198 / 5.5

t = 36 minutes

Therefore, at 6:36 p.m., the minute hand will be 3 minutes ahead of the hour hand.

Incorrect Options:

The positions of the hands for the other options do not satisfy the given condition:

6:15 p.m.:
- Minute hand is at 15 minutes (90 degrees from 12).
- Hour hand is at 6 + 15/60 = 6.25. Angle = 6.25 × 30 = 187.5 degrees from 12.
- The minute hand (90 degrees) is behind the hour hand (187.5 degrees).
6:18 p.m.:
- Minute hand is at 18 minutes (108 degrees from 12).
- Hour hand is at 6 + 18/60 = 6.3. Angle = 6.3 × 30 = 189 degrees from 12.
- The minute hand (108 degrees) is behind the hour hand (189 degrees).
6:48 p.m.:
- Minute hand is at 48 minutes (288 degrees from 12).
- Hour hand is at 6 + 48/60 = 6.8. Angle = 6.8 × 30 = 204 degrees from 12.
- The minute hand is ahead of the hour hand by 288 - 204 = 84 degrees. This angular difference corresponds to 84/6 = 14 minutes, not 3 minutes.

Thus, options A, B, and D are incorrect as they do not correspond to the minute hand being 3 minutes ahead of the hour hand.