## 5. Balls in a Box

In a box, there are 25 yellow, 42 red, 50 blue, 78 green, and 105 white balls. If we draw balls from the box without looking, how many balls do we need to draw at minimum to be sure that among the drawn balls there will be at least 50 balls of the same color?

Result: $\quad 215$
Solution.

The answer to the question in the problem is 1 greater than the answer to the following question: How many balls can we draw at most so that there are no more than 49 balls of the same color among them? And that number is equal to

$25+42+3 \cdot 49=214$.

In other words, it is possible to draw 214 balls so that there are no 49 balls of the same color among them. This will happen if we draw 25 yellow, 42 red, 49 blue, 49 green, and 49 white balls. On the other hand, if we draw 215 balls, we can be sure that we have drawn either 50 blue, 50 green, or 50 white balls.

Therefore, the answer to the question posed in the problem is:

If we draw balls from the box without looking, we need to draw at least 215 balls to be sure that there will be at least 50 balls of the same color among them.