9 x 10 x 10 x 10 = 9000
Nine is for the 1st and 7th digits. It can be anything from 1-9, just not 0 so you have 9 options. For the 2nd and 6th digits, you can have anything from 0-9 so 10 options. same for 3rd and 5th, and 4th. You don't need to do it the other way around because you'll be repeating the same numbers and you don't want that. Multiply them all together to get 9000. This is data management stuff.
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There are 900 6-digit palindromes.
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.
7 digits
There are seven (7) significant digits in 4032010.
Two significant digits, the 7 and trailing 0.