21.1.8. Reflection Method

Try these first

1. 1.

   Define a bad sequence in terms of 1s and -1s.

   Hint

   Think about what happens to the total if it dips below zero.

2. 2.

   What is the cardinality of set A?

   Hint

   Consider how many ways you can arrange n 1s and n -1s.

3. 3.

   What is a requisite condition for a sequence to be valid?

   • All sums are positive
   • All sums are non-negative
   • All sums are negative

   Hint

   Remember, we discussed partial sums' behavior extensively.

4. 4.

   True or False: The reflection method results in a sequence having more elements than the original bad sequence.

   • True
   • False

   Hint

   Think about the reflection mechanism.

5. 5.

   Given the sequence [1, 1, -1, -1, 1], identify the first negative partial sum and reflect it. What does the new sequence look like?

   Hint

   Draw the partial sums to visualize.

6. 6.

   Consider a sequence with 4 1s and 4 -1s. What would be the total valid sequences using the reflection method without calculating directly?

   Hint

   Remember the derived formula for valid sequences we discussed!