Final answer:
The configuration of friendships among 30 students where nine have three friends each, eleven have four, and ten have five is not possible because it results in a non-integer number of total friendships when properly calculated.
Step-by-step explanation:
The student is asking if it is possible for a certain configuration of friendships to exist within a class of 30 students. Specifically, whether nine students can each have three friends, eleven students can each have four friends, and ten students can each have five friends.
To answer this, we can use the concept of a handshake problem, which deals with counting the number of handshakes if each person in a group shakes hands with every other person. However, since friendships are mutual, each pairing counts only once for the entire group. Therefore, we can calculate the total friendships (or handshakes) by summing the friends each person has and then dividing by two (since each friendship involves two students).
The total number of friendships can be calculated as follows: (9 * 3) + (11 * 4) + (10 * 5) = 27 + 44 + 50 = 121 total friend connections. However, since each friendship is counted twice, we divide this by 2 to get 60.5, which is not possible since you can't have half a friendship. Therefore, it is not possible for this configuration of friendships to exist in the class.