In a math class with 24 students the individuals are polled and asked how
many members of the class they are friends with. The result of the poll is as
follows: Three 3 said they are friends with 4 persons in the class; four said they
are friends with 5 persons in the class; six said they are friends with 7 persons
in the class; seven said they are friends with 9 persons in the class; and foursaid
they are friends with 11 persons in the class Explain why some student most
have miscounted their number of friends.

Approach: This seems to be a graph theory problem, so in a graph, any node is a person, and an edge, the relationship between two persons.

All I have is that the upper bound for the # edges of a graph of 24 nodes is ${24 \choose 2} = 276$. I tried to visualize the outflow of edges in every group and found that in the group of 3 people, the outflow is 12 edges, in the group of 4, the outflow is 20, in the group of 6, the outflow is 42, in the group of 7, the outflow is 63, and in the group of 4, the outflow is 44.

Do you have in mind some theorem to solve this problem?