Counting Palindromic Numbers with Specific Conditions
<p>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:</p>
<p>A four-digit palindromic number has the form:</p>
<p>\[ abba \]</p>
<p>Where \(a\) is the first and last digit, and \(b\) is the second and third digit.</p>
<p>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 \(a\).</p>
<p>2. \(b\) can be any digit from 0 to 9, giving us 10 choices for \(b\).</p>
<p>Thus, the total number of different palindromic four-digit keys that can be generated is:</p>
<p>\[ 5 \times 10 = 50 \]</p>
<p>The answer is 50.</p>
