Question

В 22 пакета разложили конфеты трех сортов. В каждом пакете конфеты только одного сорта. Докажите, что в любом случае обязательно найдутся 8 пакетов, в которых окажутся конфеты одного сорта

Answer 1

22 разделим на 3 с остатком. получится 7 пакетов ( каждого сорта конфет) и в остатке 1 пакет. значит в один пакет мы положим конфеты одного из трех сортов, 

Answer 2

Problem Analysis

We are given 22 packages of candies, each containing candies of three different types. The task is to prove that in any case, there will always be 8 packages that contain candies of the same type.

Proof

To prove this, we can use the pigeonhole principle. The pigeonhole principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon. In this case, the pigeons are the packages and the pigeonholes are the types of candies.

Since we have 22 packages and 3 types of candies, we can divide the packages into 3 groups based on the type of candy they contain. According to the pigeonhole principle, if we have more than 3 packages in any group, then there must be at least one group with more than one package.

To find the minimum number of packages in the largest group, we can assume that each group has an equal number of packages. In this case, each group would have 22/3 = 7.33 packages. However, since we cannot have a fraction of a package, we can round up to the nearest whole number, which is 8.

Therefore, we can conclude that in any case, there will always be at least 8 packages that contain candies of the same type.

Conclusion

In any scenario where we have 22 packages of candies, each containing candies of three different types, we can prove that there will always be at least 8 packages that contain candies of the same type. This is based on the pigeonhole principle, which states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.