Question

У Пети были конфеты 7 различных видов. Он раздал их восьми девочкам. Оказалось, что конфеты одного вида он дал ровно двум девочкам. Докажите, что какой-то из девочек Петя дал конфеты только одного вида.

Answer 1

У Пети было 7 видов конфет. А девочек было 8.
Получается что 7 девочкам он дал конфеты всех 7 видов ,а одной девочке достался вид который был у одной из девочек

Answer 2

Problem Analysis

We are given that Peter (Петя) has 7 different types of candies and he distributes them to 8 girls. It is known that exactly 2 girls receive candies of the same type. We need to prove that at least one of the girls received candies of only one type.

Proof by Contradiction

To prove this, let's assume that all the girls received candies of more than one type. This means that each girl received candies of at least 2 different types. Since there are 8 girls and 7 types of candies, according to the Pigeonhole Principle, at least one type of candy must have been given to more than 2 girls.

However, this contradicts the given information that only 2 girls received candies of the same type. Therefore, our assumption that all the girls received candies of more than one type is false.

Hence, there must be at least one girl who received candies of only one type.

Conclusion

In conclusion, we have proven that at least one of the girls received candies of only one type.

Please let me know if anything is unclear or if you have any further questions!