A group of 10 students in a summer learning camp (fun times!) is asked to complete a brief survey to help them determine which of their new classmates they're most compatible with. The survey lists 10 subjects (math, science, English, art, history, band, gym, biology, shop, and home ec) and asks respondents to mark each subject as "like" or "dislike".
Once all 10 students have completed the survey, the teacher calculates an "affinity" score for each pair of students. The affinity between two students is the total number of subjects both students marked "like". Hence if Bill liked only "science", "art", "gym"; Jane liked only "science", "gym", and "history"; and Tara liked only "science", "history", "band", and "English", the affinity between Bill and Jane would be 2, the affinity between Bill and Tara would be 1, and the affinity between Jane and Tara would be 2.
The teacher writes the affinity scores for all 45 pairs of students on the board, then addresses the class.
I'm pleased to see that every student likes at least one subject, and I've noticed that no two students like exactly the same set of subjects. What I find bizarre, though, is that all of your affinity scores are even numbers. I've never seen that happen before. I'm wondering if it might be because the total number of 'likes' over all your surveys is 50 out of 100 possible; exactly half.
Hearing this, a particularly math-savvy student, Keenan Byonde, puts up his hand and explains,
Actually, it's because 50 is part of the arithmetic sequence $10 + 8n ; 0 \le n \le 10$. It's a simple matter to come up with 11 sets of ten responses to this survey such that each set has the properties:
- each respondent likes at least one subject;
- no two respondents like exactly the same subjects;
- all 45 affinity scores are even; and
- the total number of 'liked' subjects is a number in the arithmetic sequence.
Intrigued, the teacher asks Byonde to come up to the board, give an example set of ten survey responses with 50 total 'likes' and the stated properties, and explain how to (easily) contrive a set of survey responses with $10 + 8n$ total 'likes' and the stated properties for $0 \leq n \leq 10$.
What example and what simple procedure should Byonde write on the board to prove he's correct?