Final answer:
The cranial nerves responsible for moving the eyeballs are III (oculomotor), IV (trochlear), and VI (abducens), which innervate various extraocular muscles facilitating different eye movements.
Step-by-step explanation:
The cranial nerves that move the eyeballs are the oculomotor, trochlear, and abducens nerves. The oculomotor nerve (CN III) is responsible for most of the eye movements as it innervates all the extraocular muscles except the lateral rectus and superior oblique. This nerve also innervates the levator palpebrae superioris that raises the upper eyelid.
The trochlear nerve (CN IV) innervates the superior oblique muscle which enables the eye to rotate medially, also known as intorsion. Lastly, the abducens nerve (CN VI) innervates the lateral rectus muscle, which abducts the eye, pulling the eye away from the middle line of the body.
Therefore, the combination of cranial nerves responsible for eye movement is III, IV, VI.