8 Friends, 7 Tables, 1 Mixer?

📌 CONCEPT: The problem requires ensuring that no 3 friends sitting at the same table are also sitting together at any other table, which is a classic problem of permutations and combinations in graph theory.

📐 RULE / FORMULA: To solve this, we can use the concept of graph theory, specifically the idea of a 'tournament' where each pair of vertices is connected by a directed edge, representing the seating arrangement of friends.

💡 WORKED EXAMPLE: Suppose we have 8 friends (A to H) and 7 tables. We can represent the seating arrangement as a graph, where each friend is a vertex, and each edge represents a seating pair. We can then apply the rule of graph theory to ensure that no 3 friends are sitting together at any other table.

⚠️ COMMON MISTAKE: Students often get confused between permutations and combinations, thinking that the problem requires counting the number of ways to arrange the friends, when in fact it's about ensuring that no 3 friends are sitting together at any other table.