Lucky license plate numbers

First, we find the total number of possible license plate numbers which is $9999 \times 26$ .Then, we find the total number of license plate numbers which satisfy Ben's conditions.

We know that the last letter has to be W because the number is the same whether read from left to right or right to left.Then, we only need to find the number of palindromes between 1 and 9999.

All 1-digit numbers except 0 are palindromes so there are 9 1-digit palindromes.Any 2-digit palindrome must be in the form of $11 \times x$ , and there are 9 possibilities for $x$ so there are 9 2-digit palindromes.

3-digit palindromes ( in the form of $\overline{aba}$ )are a little harder. Because there are no restraints on the value of the digit in the middle ( $b$ ), there are 10 possibilities.For the other digits,there are 9 possibilities so there are $9 \times 10 = 90$ 3-digit palindromes.

4-digit palindromes are in the form of $\overline{abba}$ , where $a \neq 0$ .So,there are 99 possibilities ( 1 - 99 ) but the first 9 of them have $a = 0$ ,so they cannot be accepted.Then, we subtract to get $99 - 9 = 90$ 4-digit palindromes.

So,the chance of a random license plate number satisfying Ben's conditions is $\frac{(9 + 9 + 90 + 90)}{9999 \times 26} = \frac{198}{259974} = \frac{1}{1313}$ .So $a = 1 , b = 1313$ and $b - a = 1312$ .