Permutations & Combinations
Number formation with restricted digits
Grade 11
Question:
<p>The number of different seven-digit numbers that can be written using only the three digits 1, 2, and 3 with the condition that the digit 2 occurs twice in each number is</p>
<p>\({}^7P_2 \cdot 5^2\)</p>
<p>\({}^7C_2 \cdot 5^2\)</p>
<p>\({}^7C_2 \cdot 5^2\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Use the multinomial coefficient to distribute 7 positions among three digits where digit 2 appears exactly twice, digits 1 and 3 share the remaining 5 positions.
<p><strong>Step 1:</strong> We have 7 positions and must use digits {1, 2, 3} where digit 2 appears exactly twice.</p><p><strong>Step 2:</strong> Choose 2 positions out of 7 for digit 2: C(7,2) = 21 ways</p><p><strong>Step 3:</strong> The remaining 5 positions must be filled with digits 1 and 3. Each of these 5 positions can be filled in 2 ways (either 1 or 3).</p><p><strong>Step 4:</strong> Number of ways to fill 5 remaining positions = 2^5 = 32</p><p><strong>Step 5:</strong> Total seven-digit numbers = C(7,2) × 2^5 = 21 × 32 = 672</p><p>∴ Answer: <strong>672</strong> (Option B)</p>
Correct Answer: B