To determine how many times the hour hand and the minute hand of a clock are at right angles between 1:00 p.m. and 10:00 p.m., we need to consider the relative motion of the hands.

In a 12-hour clock, the hour hand completes one full revolution (360 degrees) in 12 hours, while the minute hand completes a full revolution in 60 minutes.

Since the minute hand moves faster than the hour hand, it covers a greater angle in the same amount of time. To find the exact positions where the hands are at right angles, we can consider each hour and calculate the angle between the two hands.

Let`s analyze the angles between the hands for each hour:

1. Between 1:00 p.m. and 2:00 p.m.:

The minute hand moves from the 12 o`clock position to the 2 o`clock position, covering 60 minutes or 360 degrees. The hour hand moves from the 1 o`clock position to the 2 o`clock position, covering 1/12th of the circle or 30 degrees. The difference in angles between the hands is 360 - 30 = 330 degrees.

2. Between 2:00 p.m. and 3:00 p.m.:

Similarly, the minute hand moves from the 2 o`clock position to the 3 o`clock position, covering 60 minutes or 360 degrees. The hour hand moves from the 2 o`clock position to the 3 o`clock position, covering 1/12th of the circle or 30 degrees. The difference in angles between the hands is again 360 - 30 = 330 degrees.

We can observe that the difference in angles between the hands remains constant for each hour. This is because both the hour hand and the minute hand move at a constant rate.

From 1:00 p.m. to 10:00 p.m., there are 10 hours. Since the difference in angles between the hands is the same for each hour, we can calculate the total number of right angles as follows:

Total number of right angles = (Number of hours) x (Number of right angles per hour)

Since there are 10 hours, the total number of right angles can be calculated as:

Total number of right angles = 10 x (330 degrees / 90 degrees) = 10 x 3.67 = 36.7

We need to round this number to the nearest whole number because we can`t have a fraction of a right angle. Therefore, the total number of right angles is 37.

Based on the options provided, none of them matches the correct answer. However, if we consider the closest option, the correct answer would be Option 3: 18.