Explanation:

- The group of 36 students has an average age of 14 years.

- The total age of all 36 students is $36\times 14=504$ years.

- Including the teacher, the group becomes 37 people with an average age of 15 years.

- The total age for 37 people is $37\times 15=555$ years.

- The teacher's age is the difference between the total age of the group with the teacher and the total age of just the students: $555-504=51$ years.

- Correct Answer: Option 4: 51