Amicable Arrangements

Why do this problem?

This problem is about arrangements and permutations, and also about logical thinking and systematic working, and presenting a solution clearly! 

There are lots of hints and suggestions within the the problem to help students along (which reduce as they question progresses).  

One of the key aspects which simplifies the problem is the symmetry of the situation.  Since this is a round table, then we can consider the "top" of the table to be wherever Elf number 1 is.  This means that we only need to consider what happens to the other 5 seats (and so we are only looking at $120$ possible arrangements rather than $720$).

Here you can find printable word and pdf versions of the problem. 

Key Questions

• What is special about a round table?  How can we use the shape of the table to simplify the problem?
• How can we label the Elves and Besties to make things clearer and simpler?
• If we know where the first elf is, in which seats can his bestie sit so that they are sitting together?  What if we want to keep them apart?

Possible Extension

Students might like to try STEP Support Programme Foundation Assignment 6, which has some further questions on arrangements and probability.

Some similar STEP questions include 1995 STEP 1 Q12, 1997 STEP 1 Q1 and 2009 STEP 1 Q13.  This can all be found via the STEP Database (you might like to type in "combinatorics" to find these quickly).