<p>There are 10 girls and 8 boys in a class room including Mr. Ravi, Ms. Rani and Ms. Radha. A list of speakers consisting of 8 girls and 6 boys has to be prepared. Mr. Ravi refuses to speak if Ms. Rani is a speaker. Ms. Rani refuses to speak if Ms. Radha is a speaker. The number of ways the list can be prepared is a 3 digit number \(n_1 n_2 n_3\), then \(|n_3 + n_2 - n_1| = \)?</p>
Step-by-Step Solution
Key Concept: Use complementary counting with constraint analysis: identify valid cases based on who speaks/doesn't speak, then apply combination principles. The constraints create mutually exclusive conditions that must be handled separately.
<p><strong>Step 1: Identify the constraints</strong><br/>• Mr. Ravi refuses if Ms. Rani speaks<br/>• Ms. Rani refuses if Ms. Radha speaks<br/>• We need 8 girls (from 10) and 6 boys (from 8)<br/>Total girls: 10 (including Ms. Rani and Ms. Radha)<br/>Total boys: 8 (including Mr. Ravi)</p><p><strong>Step 2: Partition into mutually exclusive cases</strong><br/>Let R = Mr. Ravi, Ra = Ms. Rani, Rd = Ms. Radha</p><p><strong>Case 1: Ms. Radha speaks (Rd ∈ list)</strong><br/>Then Ms. Rani cannot speak (Ra ∉ list)<br/>• Girls: Choose 7 more from remaining 8 girls: C(8,7) = 8<br/>• Boys: Choose 6 from 8 boys (R can be chosen): C(8,6) = 28<br/>• Count: 8 × 28 = 224</p><p><strong>Case 2: Ms. Radha doesn't speak, Ms. Rani speaks (Rd ∉ list, Ra ∈ list)</strong><br/>Then Mr. Ravi cannot speak (R ∉ list)<br/>• Girls: Choose 7 more from remaining 8 girls: C(8,7) = 8<br/>• Boys: Choose 6 from remaining 7 boys (excluding R): C(7,6) = 7<br/>• Count: 8 × 7 = 56</p><p><strong>Case 3: Ms. Radha doesn't speak, Ms. Rani doesn't speak (Rd ∉ list, Ra ∉ list)</strong><br/>No constraint on Mr. Ravi<br/>• Girls: Choose 8 from remaining 8 girls: C(8,8) = 1<br/>• Boys: Choose 6 from 8 boys: C(8,6) = 28<br/>• Count: 1 × 28 = 28</p><p><strong>Step 3: Calculate total number of ways</strong><br/>Total = 224 + 56 + 28 = 308</p><p><strong>Step 4: Extract digits and calculate final answer</strong><br/>308 is written as n₁n₂n₃ = 308<br/>where n₁ = 3, n₂ = 0, n₃ = 8<br/>|n₃ + n₂ - n₁| = |8 + 0 - 3| = |5| = 5<br/><br/>∴ Answer: 5</p>
Correct Answer: 5