There are 104 people in the room.
State if the following statements are true or false and
a) Explain the reasoning if true.
b) If false, give a situation where the statement is not applicable.
. . . This means provide a counter-example, proving the statement false.
1. There has to be at least two people in the room who have their birthday on the same day.
The 104 people could have 104 different birthdays.
For example: from June 1 to September 12.
2. There has to be at least 9 people who have their birthdays in February.
They could all have their birthdays in August.
3. There has to be at least 10 people who have their birthdays in the same month
Their birth-months could be distributed like this:
This is True.
4. There has to be at least 9 people who have their birthdays in the same month.
We can try to contradict the statement
. . and place only 8 people in each month:
But this accomodates only people.
There are 8 more people who have birthdays in one of the twelve months.
Wherever they are placed, it makes a month with at least 9 people.