Step 1: Use the timeline.
Panini belongs to about the 5th to 4th century BCE. Any university founded after that cannot be the answer, so we put each option on the timeline.
Step 2: Date the options.
Nalanda dates to about the 5th century CE. Valabhi rose in the 6th century CE. Vikramshila came in the 8th or 9th century CE. All three are close to a thousand years after Panini.
Step 3: Remove the late ones.
Because they came so much later, options 1, 2 and 3 are removed. Only Taxila existed in the age of Panini.
Step 4: Use geography.
Panini was from Shalatura in Gandhara, now in north-west Pakistan, close to Taxila. A young scholar from there would naturally go to the nearby centre of learning.
Step 5: Confirm.
Taxila matches both the date and the place, and it is the standard answer in textbooks, so option 4 is correct.
Final Answer:
Only Taxila fits Panini in time and in place, so option 4 is the answer.
\[ \boxed{\text{Option 4: Taxila}} \]