<p>How many odd numbers of five digits can be formed with the digits 1, 2, 3, 4, 5 if the digits cannot be repeated in the same number?</p>
Step-by-Step Solution
Key Concept: For a number to be odd, its last digit must be odd (1, 3, or 5). Fix an odd digit at the units place first, then arrange the remaining 4 digits in the remaining 4 positions.
<p><strong>Step 1:</strong> Identify odd digits available: 1, 3, 5 (three odd digits)</p><p><strong>Step 2:</strong> For a 5-digit number to be odd, the units place (last digit) must be odd.</p><p><strong>Step 3:</strong> Fix one of the 3 odd digits in the units place: <strong>3 choices</strong></p><p><strong>Step 4:</strong> Arrange the remaining 4 digits in the remaining 4 positions (ten thousands, thousands, hundreds, tens): <strong>4! = 24 ways</strong></p><p><strong>Step 5:</strong> Total odd 5-digit numbers = 3 × 24 = <strong>72</strong></p><p>∴ Answer: <strong>72</strong></p>
Correct Answer: 72