This is a classic pigeonhole principle application.
There are 12 months in a year. So assume 12 rooms, 1 room for each month. You want to put 27 students into these rooms, uniformly. How will you do that ?
- 1st round, 1 student into each room. Now each room contains 1 student, 15 students remain.
- 2nd round, 1 student again into each room. Now each room contains 2 students, 3 students remain.
- 3rd round, you can send the 3 remaining students into any of the rooms. After this, It is guaranteed that some room will have 3 students (if not more).
Hence you can conclude that there will definitely be a room with 3 students. With this analogy, you can conclude that in a group of 27 students, there will surely be at least 3 students whose birthday is in the same month.
And since the event is guaranteed to occur, probability is obviously 1.