<p>Man members of the 1st team can be selected in 15 ways and woman members of the 1st team can be selected in 15 ways. Teams are to be formed such that the number of ways decreases by 1 for each subsequent team. Find the total number of ways of selecting 15 teams.</p>
Step-by-Step Solution
Key Concept: Recognize that if men can be selected in 15 ways and women in 15 ways for team 1, then for team 2 we have 14 ways each, team 3 has 13 ways each, etc. The total ways = (15×15) × (14×14) × (13×13) × ... × (1×1) = (15!)².
<p><strong>Step 1:</strong> For Team 1: Men can be selected in 15 ways, Women can be selected in 15 ways. Total ways for Team 1 = 15 × 15 = 15²</p><p><strong>Step 2:</strong> For Team 2: Men can be selected in 14 ways, Women in 14 ways. Total ways for Team 2 = 14 × 14 = 14²</p><p><strong>Step 3:</strong> Continuing this pattern, Team k has selections = (16-k)² ways</p><p><strong>Step 4:</strong> For 15 teams total, we multiply all selections (fundamental counting principle):</p><p>Total = 15² × 14² × 13² × ... × 2² × 1²</p><p><strong>Step 5:</strong> This equals (15 × 14 × 13 × ... × 2 × 1)² = (15!)²</p><p>∴ Answer: B</p>
Correct Answer: B