7 Friends and 7 Chairs

📌 CONCEPT: The problem involves finding the number of ways to arrange the friends considering a restriction that two friends must sit together in the first two chairs.

📐 RULE / FORMULA: We can use the concept of permutations for this problem. The total number of ways to arrange the friends is given by the product of the permutations of the two 'sitting-together' friends and the remaining 5 friends, and the permutations of the 7 chairs.

💡 WORKED EXAMPLE: Consider the two friends who insist on sitting together as a single entity. There are 6! ways to arrange the remaining 6 entities (5 friends and 1 'sitting-together' entity) in 6 chairs. Within this entity, there are 2! ways to arrange the two friends. Therefore, the total number of ways to arrange the friends is 6! * 2!.

⚠️ COMMON MISTAKE: Students often forget to consider the restriction of the two friends sitting together and simply calculate the total number of permutations of 7 friends and 7 chairs, which is 7!. However, this does not take into account the restriction of the two friends sitting together.