How do I find out how many students must be in a classroom in order for at least 3 of them to have a birthday in one of January, February, March, April or at least 4 of their birthdays in one of the remaining months?
For example, if there are 3 students all with a birthday in January this fulfills the condition, however if there are 3 students, 1 with a birthday in January, 1 in February, 1 in March, this will not fulfill the condition.
You want to find the minimum number of students in the class that guarantees that either $3$ of them are born in the first $4$ months or $4$ of them are born in the last $8$ months.
Suppose there were only $12$ students in that classroom. Then there is no guarantee that the given criteria are fulfilled. It could happen that those $12$ students are born in $12$ different months. If there were $24$ students in the classroom, then again there is no guarantee of fulfilling the requirement; it could happen that exactly two of them were born in each month. Suppose there were $24+8=32$ students. Then it could happen that among them, $24$ of them are born in $12$ months with $2$ in each and the remaining $8$ are born in the months May to December, one in each. This again is not what we want. Now suppose there were $33$ students. Then the one extra student must be born either in the months Jan to April which fulfills the first criteria or born in May to December which fulfills the second criteria. So $33$ students in the class guarantees that the conditions are fulfilled.
