20.3. Parenthesizing Orders

Try these first

1. 1.

   What does C(3) represent?

   Hint

   Think about counting parentheses.

2. 2.

   State the recurrence relation for Catalan numbers.

   Hint

   It involves summing products of combinations.

3. 3.

   What is the value of C(3)?

   • 3
   • 5
   • 7

   Hint

   Count how many ways to parenthesize 4 numbers.

4. 4.

   True or False: The sequence '(()' is valid.

   • True
   • False

   Hint

   Check for unmatched pairs.

5. 5.

   Determine C(4) and explain how it relates to the sequences of valid parentheses.

   Hint

   Consider all placements of four numbers.

6. 6.

   Explain the significance of Catalan numbers in counting valid paths in combinatorics.

   Hint

   Connect the combinations to geometrical configurations.