<p><i>n</i>-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of <i>n</i> for which 900 such distinct numbers can be formed, is</p>
Step-by-Step Solution
Key Concept: Each position in an n-digit number can be filled with any of 3 digits (2, 5, or 7), giving 3^n total distinct numbers. Find the smallest n where 3^n ≥ 900.
<p><strong>Step 1:</strong> Recognize that each of the n positions in an n-digit number can independently be filled with any of 3 digits (2, 5, or 7).</p><p><strong>Step 2:</strong> By the fundamental counting principle, total distinct n-digit numbers = 3 × 3 × 3 × ... (n times) = 3^n</p><p><strong>Step 3:</strong> We need the smallest n such that 3^n ≥ 900.</p><p><strong>Step 4:</strong> Test values:</p><ul><li>3^5 = 243 (too small)</li><li>3^6 = 729 (too small)</li><li>3^7 = 2187 (sufficient, since 2187 ≥ 900)</li></ul><p><strong>Step 5:</strong> Since 3^6 = 729 < 900 but 3^7 = 2187 > 900, the smallest value of n is 7.</p><p>∴ Answer: <strong>n = 7</strong> (Option B)</p>
Correct Answer: B