This question asks us to determine which statement is necessarily true for a class of 50 students concerning their birthdays. We need to evaluate each option carefully to see if it holds true in all possible scenarios for 50 students.

Students Birthday Analysis

We have a class with 50 students. Birthdays occur throughout the year, which has 12 months. We can think of the students as 'pigeons' and the months as 'pigeonholes'. The Pigeonhole Principle is useful here. It states that if you have more items than containers, at least one container must hold more than one item. A more general form states that if you have \(n\) items to put into \(m\) containers, then at least one container must contain at least \(\lceil n/m \rceil\) items.

Evaluating Each Statement

Let's examine each statement given the 50 students and 12 months.

• Statement 1: Two students have the same birthday.

  A birthday refers to a specific day of a specific month (e.g., January 15th). There are 365 or 366 possible birthdays in a year. With only 50 students, it is possible, though highly unlikely, that all 50 students have birthdays on different days of the year. For example, if the first student is born on Jan 1st, the second on Jan 2nd, and so on, up to the 50th student born on Feb 19th (in a non-leap year), none of them would share the same birthday. Therefore, this statement is not necessarily true.

• Statement 2: Every month has birthdays of at least five students.

  If every month had at least 5 students, the total number of students would be at least \(12 \times 5 = 60\). However, there are only 50 students in the class. This distribution is impossible. It is possible for some months to have fewer than 5 students, or even zero students. Thus, this statement is not necessarily true.

• Statement 3: There exists a month which has birthdays of at least five students.

  We have 50 students (items) and 12 months (containers). Let's use the Pigeonhole Principle. To find the minimum number of students in the month with the most students, we can divide the total number of students by the number of months: \(50 / 12\).

  \[ \frac{50}{12} = 4 \text{ with a remainder of } 2 \]

  This means we can distribute the students as evenly as possible by putting 4 students in each of the 12 months, using \(12 \times 4 = 48\) students. The remaining \(50 - 48 = 2\) students must be placed into two different months. So, at least two months will have \(4 + 1 = 5\) students.

  Alternatively, using the ceiling function form of the Pigeonhole Principle, the minimum number of students in the most populated month is \(\lceil 50/12 \rceil = \lceil 4.16...\rceil = 5\). This means there must be at least one month that contains 5 or more student birthdays.

  This statement is necessarily true.

• Statement 4: The birthdays of at least 25 students are during the first six months (from January till June).

  A year is divided into 12 months. The first six months are January to June, and the last six months are July to December. We have 50 students. It is possible, for example, that all 50 students were born in the last six months of the year (July to December). In this scenario, the number of students born in the first six months would be 0, which is less than 25. Therefore, this statement is not necessarily true.

Summary of Statements

Based on the analysis:

• Statement 1 is not necessarily true.
• Statement 2 is not necessarily true.
• Statement 3 is necessarily true based on the Pigeonhole Principle.
• Statement 4 is not necessarily true.

Therefore, the only statement that is necessarily true is that there exists a month which has birthdays of at least five students.