Numerical palindromes A palindrome is an integer, such as...

Numerical palindromes A palindrome is an integer, such as 45654 , that reads the same backward and forward.
(a) How many five-digit palindromes are there?
(b) How many $n$ -digit palindromes are there?

Key Concepts

Generalization to n-Digit Palindromes

The problem can be generalized to an n-digit palindrome by identifying how many independent digits one can choose. For even n, half of the digits determine the entire number, whereas for odd n, one additional digit (the middle one) can be freely chosen. The total count is derived by multiplying the number of choices available for each independent digit, considering that the first digit cannot be zero.

Combinatorial Counting Principles

To count the total number of palindromic numbers, one uses the multiplication (or counting) principle by determining the number of independent choices available for each digit or group of digits. This approach simplifies the count by reducing the problem to considering only the independent half (or half plus one, in the case of an odd number of digits) of the digits.

Symmetry and Digit Position Constraints

In forming palindromic numbers, the digits are arranged symmetrically around a central axis. For instance, in an n-digit number, the first digit must match the last digit, the second digit must match the second-to-last digit, and so on. Additionally, the first digit cannot be zero because it would then not be a valid n-digit number. These constraints are essential for counting the valid palindromes.

Palindrome Numbers

A palindrome is a number that reads the same forwards and backwards. This symmetry means that some digits in the number are determined by others, which reduces the number of independent choices one must make when constructing such a number.

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