<p>Eight chairs are numbered 1 to 8. Two women and 3 men wish to occupy one chair each. First the women choose the chairs from amongst the chairs 1 to 4 and then men select from the remaining chairs. Find the total number of possible arrangements.</p>
Step-by-Step Solution
Key Concept: The selection process is sequential and restricted: women must choose from chairs 1-4 first, then men choose from all remaining chairs. This creates dependent events where we calculate arrangements in two separate stages.
<p><strong>Step 1:</strong> Women choose chairs from 1 to 4.</p><p>Number of ways 2 women can choose and occupy 2 chairs from chairs {1, 2, 3, 4}:</p><p>= P(4,2) = 4!/(4-2)! = 4!/2! = 4 × 3 = 12</p><p><strong>Step 2:</strong> Men choose chairs from remaining 6 chairs.</p><p>After women occupy 2 chairs, 6 chairs remain (2 from chairs 1-4 and all of chairs 5-8).</p><p>Number of ways 3 men can choose and occupy 3 chairs from these 6 remaining chairs:</p><p>= P(6,3) = 6!/(6-3)! = 6!/3! = 6 × 5 × 4 = 120</p><p><strong>Step 3:</strong> Apply multiplication principle.</p><p>Total arrangements = 12 × 120 = 1440</p><p>∴ Answer: <strong>1440</strong></p>
Correct Answer: 1440