<p>Find the three-digit odd numbers that can be formed by using the digits 1, 2, 3, 4, 5, 6 when the repetition is allowed.</p>
Step-by-Step Solution
Key Concept: For a three-digit number to be odd, it must end in an odd digit (1, 3, or 5). Use the multiplication principle by fixing the units place first, then counting choices for hundreds and tens places independently.
<p><strong>Step 1:</strong> Identify the structure of a three-digit odd number.</p><p>For a number to be odd, the units digit (rightmost position) must be odd.</p><p>Available odd digits from {1, 2, 3, 4, 5, 6}: {1, 3, 5}</p><p><strong>Step 2:</strong> Count choices for each position using the multiplication principle.</p><p>• <strong>Units place (ones):</strong> Must be odd → 3 choices {1, 3, 5}</p><p>• <strong>Hundreds place:</strong> Can be any digit from {1, 2, 3, 4, 5, 6} → 6 choices (repetition allowed)</p><p>• <strong>Tens place:</strong> Can be any digit from {1, 2, 3, 4, 5, 6} → 6 choices (repetition allowed)</p><p><strong>Step 3:</strong> Apply multiplication principle.</p><p>Total three-digit odd numbers = 6 × 6 × 3 = 108</p><p>∴ Answer: <strong>108</strong></p>
Correct Answer: 108