22.2.1. Counting Solutions to an Equation

Try these first

1. 1.

   Define the Principle of Inclusion-Exclusion.

   Hint

   Think of how two sets might overlap.

2. 2.

   What does the cardinality of a set represent?

   Hint

   It’s a measure of size!

3. 3.

   What does the inclusion-exclusion principle help us calculate?

   • The length of sets
   • The area of shapes
   • The number of elements in unions of sets

   Hint

   Think about what unions and overlaps imply.

4. 4.

   Is it true that a derangement can have elements in their original positions?

   • True
   • False

   Hint

   Remember the definition of a derangement!

5. 5.

   Calculate the number of distinct derangements for 5 objects using PIE.

   Hint

   Calculate permutations accounting for intersections carefully.

6. 6.

   How many onto functions exist from a set with 8 elements to one with 4 elements?

   Hint

   Define your subsets explicitly for clarity!