A number is called a palindrome if it reads the same backward as...

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

Numbers can be between $0 - 9$

Number of ways of selecting value of $a = P 110$

Similarly number of ways of choosing value of $b = P 110$ (As a and b can also be same)

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