Question

"A palindrome is a sequence of characters which reads the same forward and

backwards like RACECAR or 10011011001.
(A) How many bit strings of length 10 are palindromes?

Answer 1

There are 2^5 = 32 bit strings of length 10 that are palindromes.

Explanation:

A palindrome is a sequence of characters that reads the same forward and backward. For bit strings of length 10, we can exploit the symmetry of palindromes to simplify the counting process. Since the first half uniquely determines the second half in a palindrome, we only need to consider the first 5 bits.

For each of these 5 bits, there are two possibilities: 0 or 1. Thus, the total number of palindromic bit strings of length 10 is 2^5, which equals 32. This is because, for each choice of the first 5 bits, the second half is uniquely determined by mirroring the choices made for the first half.

In summary, the answer to the question is that there are 32 bit strings of length 10 that are palindromes. This result is derived from understanding the inherent symmetry in palindromes, which simplifies the counting process by considering only the first half of the bit string.

Answer 2

There are 32 different palindromic bit strings of length 10, because only the first five bits of the string need to be specified.

Step-by-step explanation:

The question is asking for the number of bit strings of length 10 that are palindromes. A palindromic bit string is one that reads the same forwards and backwards, much like palindromic sequences in DNA. To calculate this, we need to consider that for a bit string of length 10, only the first five bits need to be specified, because the last five bits will be the mirror image of the first five, to maintain the palindrome property.

To find how many 10-bit palindromic strings exist, we can choose any combination of 0 or 1 for each of the first five positions. Since there are two possibilities for each position, and five positions that can be independently chosen, the total number of 10-bit palindromic strings is 2^5, which is 32.

Therefore, there are 32 different palindromic bit strings of length 10.