20.2.1. Understanding C(n)

Try these first

1. 1.

   What is C(2)?

   Hint

   Think of the arrangements for three numbers.

2. 2.

   How many valid strings can be created with 1 pair of parentheses?

   Hint

   Count the number of valid pairs.

3. 3.

   What does C(3) represent?

   • Ways to parenthesize 3 numbers
   • Ways to parenthesize 4 numbers
   • Ways to parenthesize 5 numbers

   Hint

   Remember the definition of C(n).

4. 4.

   True or False: C(n) is an example of a combinatorial sequence.

   • True
   • False

   Hint

   Think about what combinatorial problems involve.

5. 5.

   Calculate C(5) using the recurrence relation.

   Hint

   Use the known values of C(0)-C(4) to calculate.

6. 6.

   Show that the number of valid parentheses of length 2n is equal to C(n).

   Hint

   Draw connections between parentheses arrangement and multiplication.