Between 5 o'clock and 7 o'clock, the minute hand and the hour hand of a clock will be opposite to each other only once. Let's break down the explanation into different sections:

Understanding the clock:
1. A clock has 12 numbers on its dial, representing the hours.
2. The minute hand of a clock completes a full revolution in 60 minutes, while the hour hand takes 12 hours to complete a full revolution.
3. The minute hand moves 12 times faster than the hour hand because it covers the same distance in 60 minutes that the hour hand covers in 12 hours.

Calculating the positions:
1. At 5 o'clock, the hour hand will be pointing exactly at the 5, while the minute hand will be at the 12.
2. As time progresses, the hour hand moves slowly towards the next hour, while the minute hand moves around the dial.
3. At 6 o'clock, the hour hand will be pointing at the 6, while the minute hand will be at the 12 again.
4. The minute hand will have made a complete revolution around the dial, crossing the hour hand three times during this period.

Determining the opposite positions:
1. The opposite positions occur when the minute hand and the hour hand are exactly on opposite sides of the clock.
2. In other words, the minute hand is at the 6 while the hour hand is at the 12, or vice versa.
3. At 5 o'clock, the minute hand is not exactly on the 6, so it is not opposite to the hour hand.
4. At 6 o'clock, the minute hand is exactly on the 6, while the hour hand is at the 12, making them opposite to each other.
5. Therefore, between 5 o'clock and 7 o'clock, the minute hand and the hour hand come opposite to each other only once.

Conclusion:
Between 5 o'clock and 7 o'clock, the minute hand and the hour hand of a clock are opposite to each other only once. Therefore, the correct answer is option 'C'.