Answer:
The number of palindromes is
[tex] 2^{\frac{n}{2} }[/tex] when n is even and
[tex]2^{\frac{n +1}{2} }[/tex] when n is odd
Step-by-step explanation:
From the question we are told that
The length of the string is [tex]n[/tex]
Generally palindrome is evaluated by considering the first part of a string
When the the length of the string is an even number then
it means that the first part of the string is [tex]\frac{n}{2}[/tex]
Hence the number of bit strings of length n that are palindromes is evaluated as
[tex]p(n_{even }) = 2^{\frac{n}{2} }[/tex]
But When the the length of the string is an odd number then
it means that the first part of the string is [tex]\frac{n-1}{2}[/tex]
Hence the number of bit strings of length n that are palindromes is evaluated as
[tex]p(n_{odd }) =2^{\frac{n -1}{2} +1 } = 2^{\frac{n +1}{2} }[/tex]
Generally each bit could be either 0 or 1
Hence the number of palindromes is
[tex] 2^{\frac{n}{2} }[/tex] when n is even and
[tex]2^{\frac{n +1}{2} }[/tex] when n is odd
