Student Birthdays
A
math teacher says:
I am certain that at least 6 students were born on the same day of the
week, but I am not sure if there are 4 who were born in the same month.
Question:
How many students are in the class?
Answer:
To determine the number of students in the class, we can use the
pigeonhole principle (or Dirichlet's principle).
1. For the days
of the week:
- There are 7 days in a week.
- If at least 6 students were born on the same day
of the week, this means there are at least 6 students for each day of
the week.
- Therefore, to be certain that at least 6
students were born on the same day of the week, there must be at least
(7 x 6 = 42)
students.
2. For the months of the year:
- There are 12 months in a year.
- If we are not sure that there are 4 students
born in the same month, this means there could be at most 3 students
born each month.
- Therefore, to be certain that there are not 4
students born in the same month, there must be at most (12 x 3 = 36) students.
To satisfy both conditions, the number of students must be between 37
and 41. However, since we know that at least 6 students were born on
the same day of the week, the minimum number of students must be 42.
Therefore, there are 42 students in the class.
