2013 AMC 12A Problems/Problem 22
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome is chosen uniformly at random. What is the probability that is also a palindrome?
By working backwards, we can multiply 5-digit palindromes by , giving a 6-digit palindrome:
Note that if or , then the symmetry will be broken by carried 1s
Simply count the combinations of for which and
implies possible (0 through 8), for each of which there are possible C, respectively. There are valid palindromes when
implies possible (0 through 7), for each of which there are possible C, respectively. There are valid palindromes when
Following this pattern, the total is
6-digit palindromes are of the form , and the first digit cannot be a zero, so there are combinations of
So, the probability is
You can more easily count the number of triples by noticing that there are possible values for and possible values for once is chosen. Summing over all , the number is By the hockey-stick identity, it is .
Solution 2 (using the answer choices)
Let the palindrome be the form in the previous solution which is . It doesn't matter what is because it only affects the middle digit. There are ways to pick and , and the only answer choice with denominator a factor of is .
Video Solution by Richard Rusczyk
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