Hello, carriereg!

Quote:

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.

Quote:

1. There has to be at least two people in the room who have their birthday on the same day.

False.

The 104 people could have 104 *different* birthdays.

For example: from June 1 to September 12.

Quote:

2. There has to be at least 9 people who have their birthdays in February.

False.

They could all have their birthdays in August.

Quote:

3. There has to be at least 10 people who have their birthdays in the same month

False.

Their birth-months could be distributed like this:

. . $\displaystyle \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline

\text{Jan} & \text{Feb} & \text{Mar} & \text{Apr} & \text{May} & \text{Jun} & \text{Jul} & \text{Aug} & \text{Sep} & \text{Oct} & \text{Nov} & \text{Dec} \\ \hline

9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 5 \\ \hline

\end{array}$

Quote:

4. There has to be at least 9 people who have their birthdays in the same month.

This is *True*.

We can *try* to contradict the statement

. . and place only 8 people in each month:

. . $\displaystyle \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline

\text{Jan} & \text{Feb} & \text{Mar} & \text{Apr} & \text{May} & \text{Jun} & \text{Jul} & \text{Aug} & \text{Sep} & \text{Oct} & \text{Nov} & \text{Dec} \\ \hline

8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\ \hline

\end{array}$

But this accomodates only $\displaystyle 8 \times 12 \:=\:96$ 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.