12.3. Pascal's Identity

Try these first

1. 1.

   What is a combinatorial proof?

   Hint

   Think about counting instead of algebra.

2. 2.

   State Pascal's Identity.

   Hint

   It relates binomial coefficients in a specific way.

3. 3.

   What does Pascal's Identity state?

   • C(n
   • k) = C(n - 1
   • k) + C(n - 1
   • k - 1)
   • C(n
   • k) = C(n + 1
   • k) - C(n - 1
   • k)
   • C(n
   • k) = C(n
   • n - k)

   Hint

   Remember, it involves choosing with or without a specific item.

4. 4.

   True or False: A combinatorial proof involves simplifying algebraic expressions.

   • True
   • False

   Hint

   Think about what a counting argument means.

5. 5.

   Using combinatorial proof, demonstrate that C(n, k-1) + C(n, k) = C(n+1, k).

   Hint

   Think of including one extra item in the group.

6. 6.

   Find the total number of ways to form teams of 4 from 10 members using Pascal's Identity.

   Hint

   Consider the incremental selections through Pascal's layers.