12.1. Definition of Combinatorial Proofs

Try these first

1. 1.

   What is a combinatorial proof?

   Hint

   Think about how you count objects in different ways.

2. 2.

   What does LHS stand for?

   Hint

   It refers to the left part of an equation.

3. 3.

   What does a combinatorial proof primarily rely on?

   • Algebraic manipulation
   • Counting arguments
   • Geometric proofs

   Hint

   Remember the essence of combinatorial reasoning.

4. 4.

   True or False: Combinatorial proofs can involve expanding expressions.

   • True
   • False

   Hint

   Think about the method's purpose.

5. 5.

   Using combinatorial proof, establish the identity C(n, k-1) + C(n, k) = C(n+1, k).

   Hint

   Break the problem into cases based on selection.

6. 6.

   Create a combinatorial proof for the expression C(n, r) = C(n-1, r) + C(n-1, r-1).

   Hint

   Think about selections where the specific object is part of 'r' or not.