23.5. Derangements of n Objects

Try these first

1. 1.

   What is a derangement?

   Hint

   Think about arrangements that avoid placing objects back to where they started.

2. 2.

   Provide an example of a derangement for the set {1, 2}.

   Hint

   Consider the fact that there are only two objects.

3. 3.

   What is the definition of a derangement?

   Hint

   Recall the meaning of arrangements.

4. 4.

   True or False: D(1) = 1.

   • True
   • False

   Hint

   Consider the number of ways to arrange one object.

5. 5.

   Given 5 distinct objects, how many derangements can be formed? Show your calculations.

   Hint

   Use the established recurrence and substitute known values.

6. 6.

   Explore a new configuration where no object can be in its direct neighbor’s position (e.g. a line of people). What changes?

   Hint

   Think about the implications of forbidding adjacent positions.