21.1.12. Counting 1s and -1s in S'

Try these first

1. 1.

   What does a valid sequence of 1s and -1s imply?

   Hint

   Think about summation and how it behaves.

2. 2.

   Calculate C(4, 2).

   Hint

   This is a simple combinatorial count.

3. 3.

   What is the formula for the total number of sequences of n 1s and -1s?

   • C(2n
   • n)
   • C(2n
   • n+1)
   • C(2n
   • 0)

   Hint

   This is basic combinatorics — think pairs!

4. 4.

   True or False: Bad sequences maintain non-negative sums.

   • True
   • False

   Hint

   Reflect on our definition of bad sequences.

5. 5.

   Consider the sequence S = {1, -1, 1, -1, -1, 1}. Show it's a bad sequence and demonstrate what its reflection would look like.

   Hint

   Trace through the negative point.

6. 6.

   Find the number of valid sequences for n=4 and explain why they are equivalent to C(8, 4) - C(8, 5).

   Hint

   Break it down into visual paths.