5. At the end of the school year, Principal Srećko conducted a survey among students about losing personal items. Out of 1000 students in his school, each has lost at least one of three items this school year: a notebook, keys, or a pen. 700 of them lost a notebook, 750 lost keys, and 800 lost a pen.

How many students at least lost all three items this school year?

## Comment before the solution.

The answer to the question is 250. Of course, the answer needs to be justified.

In this case, the justification consists of two parts. It is necessary to:

- prove that it is not possible for fewer than 250 students to lose all three items and
- provide an example showing that it is indeed possible for exactly 250 students to lose all three items.

Note that each of the following solutions includes these two parts.
## First Solution.

If 700 students in the school have lost their notebooks, then 300 have not lost their notebooks (colored yellow).

If 750 students in the school have lost their keys, then 250 have not lost their keys (blue).

If 800 students in the school have lost their pens, then 200 have not lost their pens (purple). Let $x$ be the number of students who have lost all three items (green).
![](https://cdn.mathpix.com/cropped/2024_05_30_38ebdf2cda2a1fe1ae2eg-6.jpg?height=996&width=592&top_left_y=586&top_left_x=210)
![](https://cdn.mathpix.com/cropped/2024_05_30_38ebdf2cda2a1fe1ae2eg-6.jpg?height=444&width=1108&top_left_y=578&top_left_x=777)

These four sets (colored yellow, blue, purple, and green) together cover all students, as there is no student who has not lost any of the notebook, keys, or pen. Therefore, the sum of the number of students in these four sets is greater than or equal to the total number of students in the school, so we have:

$$
\begin{gathered}
300+250+200+x \geq 1000 \\
750+x \geq 1000 \\
x \geq 250
\end{gathered}
$$

This means that the smallest possible number of students who have lost all three items is greater than or equal to 250.

Let's show by example that this number is exactly 250.

![](https://cdn.mathpix.com/cropped/2024_05_30_38ebdf2cda2a1fe1ae2eg-6.jpg?height=415&width=1529&top_left_y=1793&top_left_x=233)

Here, 250 students who have lost all three items are schematically represented, 300 students who have lost their pens and keys, 250 students who have lost their pens and notebooks, and 200 students who have lost their keys and notebooks.

This means that it is possible for exactly 250 students to have lost all three items.