How many 13 digit numbers are palindromes and have at least one place where a digit is doubled. A palindrome is the same read forwards or backwards, like bob, racecar, or 57975.

For example:
1234456544321
9107772777019
2497644467942

1234567654321 Does not count, since it has no doubled digits.

I need help with how to solve....Thanks and sorry if this is not advanced probability

2. Originally Posted by tony351
How many 13 digit numbers are palindromes and have at least one place where a digit is doubled. Answer: 4217031
I assume that a 13 digit number cannot begin with a zero.
The create a 13 digit palindrome all we need is to create a seven digit number and reverse the order of the first six digits.
Thus there are $\displaystyle 9\left( {10^6 } \right)$ such palindromes.
To count the number of those with no repeated digits, each time we select a digit we must select a different one the next time. That can be done in $\displaystyle 9\left( {9^6 } \right)$ possible ways.
So the answer: $\displaystyle 9\left( {10^6 } \right) - 9\left( {9^6 } \right)$.

3. ## stupid question

Thank you so much for the solution, can you tell me where the 9 comes from? I know that's a crazy question but I have no clue. Thanks again in advance

4. Originally Posted by tony351
Can you tell me where the 9 comes from?
Simple. The number cannot began with a zero.
Thus that are but nine choices for the first digit in the number.

5. ## thank you

See told you dumb question LOL. Thank you it makes since now