Palindrome Numbers only in this form will be divisible by $55$.

Numbers can be  between $0-9$

Number of ways of selecting value of $a={}_1{}^{10}P$

Similarly  number of ways of choosing value of $b={}_1{}^{10}P$  (As a and b can also be same)

Therefore, Total number of ways of getting palindrome numbers divisible by $55$ is $=1\times 10\times 10\times 1=100$