21.1.3. Second Problem: Sequences of 1s and -1s

Try these first

1. 1.

   How many sequences are formed with 2 1s and 2 -1s?

   Hint

   Think about choosing positions for 1s.

2. 2.

   If a sequence has more -1s than 1s, is it valid?

   Hint

   Recall the definition of valid sequences.

3. 3.

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

   • C(2n
   • n)
   • C(2n
   • n + 1)
   • Both A and B

   Hint

   Think of how we derive valid counts.

4. 4.

   True or False: If a sequence has an initial position with a negative sum, it is invalid.

   • True
   • False

   Hint

   Refer back to the definition of valid sequences.

5. 5.

   Find the number of valid sequences for n=4 and explain your approach.

   Hint

   Make sure to outline both sets clearly.

6. 6.

   Create a sequence of n=5 and justify its validity.

   Hint

   Visualize how the sum changes as you construct the sequence step-by-step.