Problem 6.6. Four girls and eight boys came to take a class photo. The children approach the photographer in pairs and take a joint photo. Among what minimum number of
photos will there definitely be either a photo of two boys, or a photo of two girls, or two photos of the same children?
Answer: 33.

Solution. Suppose at some point there is neither a photo of two boys, nor a photo of two girls, nor two photos of the same children. Then, on each photo, there is a boy and a girl, and on different photos, there are different pairs. However, the total number of possible pairs consisting of a boy and a girl is $4 \cdot 8=32$, and each of these pairs can be captured in no more than one photo. Therefore, among any 33 photos, there must be either a photo of people of the same gender, or two photos of the same pair of people.

At the same time, 32 photos may not be enough: they can capture different pairs consisting of a boy and a girl.