109
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1999
A palindrome reads the same forwards as backwards. Therefore the first and last digits must be the same, the second and penultimate the same, and so on. As 1000 & 3000 both have 4 digits, there are two pairs of digits that must be the same: the first and fourth, the second and third. The first digit (and hence last digit) can only be 1 or 2 - a choice of 2 The second digit (and hence third digit) can be any of 0, 1, 2, ..., 9 - a choice of 10 for each of the choices of the first/last digit. Thus there are 2 × 10 = 20 palindromes between 1000 and 3000.
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999 of them.