The hands of a clock coincide 22 times in a day — option 4.

The relative speeds. The minute hand completes 360° in 60 minutes, so it moves at 6° per minute. The hour hand completes 360° in 12 hours, so it moves at 0·5° per minute. The minute hand therefore gains on the hour hand at

\(6-0.5=5.5\ \text{degrees per minute}\)

For the hands to coincide again after a coincidence, the minute hand must gain a full 360°, which takes

\(\dfrac{360}{5.5}=\dfrac{720}{11}=65\dfrac{5}{11}\ \text{minutes}\)

Counting the coincidences. A day has 1,440 minutes, so the number of gains of a full circle is

\(\dfrac{1440}{720/11}=\dfrac{1440\times11}{720}=22\)

Why it is 22 and not 24 — the point of the question. One might expect the hands to meet once an hour, which would give 24 coincidences. They do not, because each meeting takes about 65⁵⁄₁₁ minutes rather than 60, so the meetings slip steadily later. In each 12-hour cycle there are only 11 coincidences, and the hour between 11 o’clock and 12 o’clock has none: the hands that would meet at “11:60” in fact meet exactly at 12:00, which is counted in the next cycle. So 11 per half day gives 22 per day.

The meeting times in a 12-hour cycle are 12:00, then approximately 1:05:27, 2:10:55, 3:16:22, and so on at intervals of 65⁵⁄₁₁ minutes, ending with the meeting at 12:00 again.

Hence, the answer is 22.