12.3.1. Category 1 of Combinations

Try these first

1. 1.

   What is a combinatorial proof?

   Hint

   Think about how we count objects.

2. 2.

   Give an example of a combinatorial identity.

   Hint

   Consider how combinations link together.

3. 3.

   What is the goal of a combinatorial proof?

   • To simplify expressions
   • To prove counts are equal
   • To expand formulas

   Hint

   Remember that simplification is not involved!

4. 4.

   True or False: In combinatorial proofs, you often simplify the expressions involved.

   • True
   • False

   Hint

   Focus on the nature of proofs we've discussed.

5. 5.

   Prove $C(n, k) = C(n, n-k)$ using a combinatorial proof technique. What are the two perspectives?

   Hint

   Reflect upon how could view the same set in two different ways.

6. 6.

   Use Pascal's identity to demonstrate a relationship involving $C(5, 3)$. What are the distinct cases?

   Hint

   Consider the choices available based on including or excluding one specified element.