Detailed Explanation:

The hour and minute hands of a clock coincide when the angle between them is 0°. At 5 o'clock, the hour hand is at 150° (30° × 5). The minute hand moves at 6° per minute, while the hour hand moves at 0.5° per minute, creating a relative speed of 5.5° per minute.

To close the 150° gap: M = 150 ÷ 5.5 = 300/11 = 27³⁄₁₁ minutes. This can also be derived using the standard formula for hand coincidence: M = (60H)/11, where H = 5.

Therefore, the hands coincide at 27³⁄₁₁ minutes past 5, making Option 2 the correct answer.