On Binary Island, the locals have only two letters in their alphabet:

Question

A and B. Sequences of these letters are called strings. The number of
letters in a string is called its length. If a string has length n, then we
call it an n-string. For example, ABAABABAABAA is a 12-string.
A block of consecutive letters in a string is called a substring. A substring
may appear more than once. For example, ABAABABAABAA contains
the substring AA three times, the substrings AB and BA four times each,
but does not contain the substring BB.
A string that reads the same forwards and backwards is called palindromic. Every 3-string starting with A has exactly three different palindromic substrings, as shown in this table.

3-string palindromic substrings
AAA -> A, AA, AAA
AAB -> A, B, AA
ABA -> A, B, ABA
ABB -> A, B, BB

a) Explain why it follows from the table above that every 3-string starting with B has exactly 3 palindromic substrings.
b) Show that every 4-string starting with A has exactly 4 palindromic
substrings.
c) Show that every 5-string has exactly 5 palindromic substrings.
It is also true that every 6-string has exactly 6 palindromic substrings,
and every 7-string has exactly 7 palindromic substrings. However, this
pattern does not continue.
d) Find an 8-string that starts with AABBA and has only 7 palindromic
substrings.