There are 11520 of them.
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90000. With 10 digit palindromes, the last 5 digits are the same as the first 5 digits in reverse, eg 12345 54321. So it comes down to how many 5 digit numbers are there? They are the numbers "10000" to "99999", a total of 99999 - 10000 + 1 = 90000.
There are 900 three digit numbers between 99 and 3,000 - ranging from 100 to 999
271 of the first 1000 natural numbers contain at least one digit 5. That is 27.1 % of them.
If you include the number 3,000, there are exactly 2,001 4-digit numbers between 500 and 3000.