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Question

Consider a sequence of letters where each letter is A or B. We can call the sequence stable if, when we tally the numbers of As and the number of Bs in the sequence working from left to right, the difference is never greater than one.

For example, the sequence ABBABA is stable but the sequence AABBAB is not, because after counting the first two letters, the difference is two.

How many stable sequences with eighteen letters are there?

Answer 1

Answer: There are 512 stable sequences of eighteen letters.

Explanation:

1. Stability Requirement: A sequence of As and Bs is called stable if, when assigning a score of +1 for A and –1 for B, the running total (or balance) after reading each letter always satisfies -1<= x <= 1. The difference between the number of As and Bs never exceeds 1 in absolute value.
2. Example analysis (Consider the sample sequence ABBABA)

First letter:

Choosing A gives a score of +1, and choosing B gives –1. Both choices are valid since ±1. This first step is a free choice.

Second letter:

If the first letter is A (score +1), adding A again would yield a score of +2, which is not allowed. Thus, the second letter must be B, forcing the score back to 0. Similarly, if the first letter were B (score –1), the second letter must be A.

Third letter:

Once the running total is 0 again (after two letters), you have a free choice for the third letter. No matter whether you choose A (score becomes +1) or B (score becomes –1), the condition remains satisfied.

Continuing this Pattern:

Every time the running total is 0 (which happens after every second letter), you have a free choice for the next letter. However, once you choose an A or B and move to a running total of +1 or –1, the next letter is forced (to avoid exceeding the limit), as it must bring the total back to 0.

Counting the Free Choices:

Since the total sequence length is 18 (an even number), the letters can be grouped into 9 pairs. In each pair:

1. The first letter (odd-numbered position) is a free choice (2 possibilities: A or B).
2. The second letter (even-numbered position) is forced, as it must offset the running total back to 0.

Thus, the total number of stable sequences is given by:

2⁹ = 512