23.5.2. Overall Formula for Derangements

Try these first

1. 1.

   List all derangements for 3 elements A, B, C.

   Hint

   Recall that no element can be in its original position.

2. 2.

   What is the definition of a derangement?

   Hint

   Think about how this affects their placements.

3. 3.

   What defines a derangement?

   Hint

   Think about what happens to items in a shuffled arrangement.

4. 4.

   True or False: The formula for derangements is D(n) = (n - 1) * (D(n - 1) + D(n - 2)).

   Hint

   This formula shows how previous results can define new ones.

5. 5.

   Prove that D(n) = (n!)(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!) using induction.

   Hint

   Recall how factorial terms alter when establishing this equation.

6. 6.

   How do derangements apply in generating random matches in a class?

   Hint

   This can help in discussions and keeping partnerships dynamic.