Question

How many bit strings of length 12 are palindromes and contain 10 as sub string

Answer 1

There are as many palindromic 12-bit binary strings as there are permutations with repetition of 6-bit binary strings, which is equal to $2^6=64$.

There are also only two permutations with repetition of 6-bit binary strings, that when each of them is mirrored and the result is "glued" to the right side of the original string, will give a palindromic string without 10 as a substring.
Those strings are 000000 and 111111.

Therefore there are $64-2=62$ palindromic 12-bit binary strings with 10 as substring.