Question - Counting Palindromic Numbers with Specific Conditions

To find the number of different palindromic keys that can be generated using four-digit numbers with the specified conditions (even, odd, and four digits), consider the following:

A four-digit palindromic number has the form:

\[ abba \]

Where \(a\) is the first and last digit, and \(b\) is the second and third digit.

1. \(a\) must be an odd digit (1, 3, 5, 7, 9) to satisfy the condition of being odd. Therefore, there are 5 choices for \(a\).

2. \(b\) can be any digit from 0 to 9, giving us 10 choices for \(b\).

Thus, the total number of different palindromic four-digit keys that can be generated is:

\[ 5 \times 10 = 50 \]

The answer is 50.