The correct answer to the question is option 4, which is 11.

To determine the number of times the hands of a watch are at a right angle between 4 p.m. and 10 p.m., we need to consider the positions of the hour and minute hands at different times.

At 4 p.m., the hour hand points directly at the 4 and the minute hand points at the 12. These hands are not at a right angle.

As time progresses, the minute hand moves around the clock at a faster pace than the hour hand. At 5 p.m., the minute hand would have moved away from the hour hand, forming a smaller angle.

The hands will be at a right angle when the minute hand is directly on one of the hour marks (1, 2, 4, 5, 7, 8, 10, and 11), while the hour hand is on the opposite side of the clock face.

From 4 p.m. to 10 p.m., there are 6 hour marks that the minute hand will align with while the hour hand is on the opposite side. This gives us 6 instances.

However, between 4 p.m. and 5 p