23.5.1. Categories of Derangements

Try these first

1. 1.

   What is a derangement?

   Hint

   Think of a simple example with three objects.

2. 2.

   Provide a value for D(0).

   Hint

   Consider the definition of derangement.

3. 3.

   What defines a derangement?

   • All elements remain in their place
   • At least one element is out of place
   • No element remains in its original position

   Hint

   Think about the definition.

4. 4.

   Is D(2) greater than D(1)?

   • True
   • False

   Hint

   Compare the values calculated.

5. 5.

   For n = 6, challenge yourself by calculating D(6) using both values from the recurrence relation and directly enumerating possibilities.

   Hint

   Use previous calculations to help.

6. 6.

   Create a graphical representation of the derangements for n=4. Illustrate the invalid placements to better visualize.

   Hint

   Think about arranging four objects and sketching their positions.