<p>Total number less than \(3 \times 10^8\) and can be formed using the digits 1, 2, 3 is equal to</p>
<p>(1) \(\dfrac{1}{2}(3^9 + 4 \times 3^8)\)</p>
<p>(2) \(\dfrac{1}{2}(3^9 - 3)\)</p>
<p>(3) \(\dfrac{1}{2}(7 \times 3^8 - 3)\)</p>
<p>(4) \(\dfrac{1}{2}(3^9 - 3 + 3^8)\)</p>
Step-by-Step Solution
Key Concept: Count numbers by digit length (1-digit, 2-digit, 3-digit) where 3-digit numbers must start with 1 or 2 to stay below 3×10⁸. Since 3×10⁸ has 9 digits, all numbers up to 8 digits automatically satisfy the constraint.
<p><strong>Step 1:</strong> Identify the constraint. Numbers must be less than 3×10⁸ = 300,000,000 using digits {1, 2, 3}.</p><p><strong>Step 2:</strong> Count by number of digits (repetition allowed):</p><p>• 1-digit numbers: 3¹ = 3</p><p>• 2-digit numbers: 3² = 9</p><p>• 3-digit numbers: 3³ = 27</p><p>• 4-digit numbers: 3⁴ = 81</p><p>• 5-digit numbers: 3⁵ = 243</p><p>• 6-digit numbers: 3⁶ = 729</p><p>• 7-digit numbers: 3⁷ = 2,187</p><p>• 8-digit numbers: 3⁸ = 6,561</p><p><strong>Step 3:</strong> For 9-digit numbers starting with 1 or 2: 2 × 3⁸ = 2 × 6,561 = 13,122</p><p><strong>Step 4:</strong> Sum all valid numbers:</p><p>3 + 9 + 27 + 81 + 243 + 729 + 2,187 + 6,561 + 13,122 = 22,962</p><p><strong>Alternate formula:</strong> (3⁹ - 3⁸)/2 + (3⁸ - 1)/2 = 13,122 + 9,840 = 23,962 or using geometric series: Total ≈ 29,524 (verify with option C)</p><p>∴ Answer: C</p>
Correct Answer: C