## Task A-1.3.

In the box, there are $k$ balls labeled (k) for all $k=1,2, \ldots, 50$ (thus one ball labeled (1), two balls labeled (2), ..., 50 balls labeled (50).

Balls are drawn from the box without looking. How many balls must be drawn to ensure that at least 10 balls with the same label have been drawn?
## Solution.

The answer to the question in the task is 1 greater than the answer to the following question:

How many balls can we draw (looking at them, i.e., choosing which balls to draw) so that among them, there are no more than 9 balls with the same label?

We will achieve this by taking nine balls with each label (or fewer if there are not enough).

![](https://cdn.mathpix.com/cropped/2024_05_30_acf83088931d085246bdg-02.jpg?height=845&width=1350&top_left_y=1782&top_left_x=296)

The total number of such balls is

$$
\begin{aligned}
1+2+3+4+5+6+7+8+9+9+9+\cdots+9 & =(1+2+\cdots+8)+42 \cdot 9 \\
& =36+378=414
\end{aligned}
$$

Thus, it is possible to draw 414 balls such that there are no ten balls with the same label among them.

On the other hand, if we draw 415 balls, we can be sure that we have drawn ten balls with one of the labels (10), (11), (12), ..., (50).

We need to draw at least 415 balls to be sure that among the drawn balls, at least ten balls have the same label.

Note: Above the line are the labels of the balls that we can draw so that there are no 10 with the same label. Counting by rows, we can also see that their number is $50+49+\cdots+42=414$.