Step 1: List who supplies the ear canal.
Several nerves share the skin of the ear canal: the trigeminal nerve up front, the vagus nerve mostly at the back and floor, and small patches from the facial and glossopharyngeal nerves.
Step 2: Trace where the vagus nerve goes next.
The same vagus nerve carries sensation from the throat, voice box, and airway, the exact spots that set off a cough when irritated. Its ear branch, called Arnold's nerve, taps into this same wiring.
Step 3: Put the two facts together.
So when a cotton bud or water touches the vagus-supplied skin in the canal, the brain reads it as if the airway were being irritated, and a cough follows. This is Arnold's reflex, sometimes called the ear-cough reflex.
Step 4: Check the distractors.
The trigeminal nerve and the small facial nerve twigs feel the canal too but do not connect to the cough pathway. C1 to C2 fibres via the great auricular nerve serve the outer ear skin, not the canal's cough trigger.
Step 5: Give the answer.
$\text{The X cranial nerve}$ (vagus), through Arnold's nerve, is behind this cough response.
\[ \boxed{\text{The X cranial nerve}} \]