<p>Eleven animals of a circus have to be placed in eleven cages (one in each cage). If 4 of the cages are too small for 6 of the animals, find the number of the ways of caging all the animals.</p>
Step-by-Step Solution
Key Concept: Partition the problem into two groups: 6 large animals must avoid 4 small cages (they go to 7 large cages), and 5 remaining animals fill the leftover positions. Use conditional placement rather than direct arrangement.
<p><strong>Step 1:</strong> Identify the constraints. We have 11 animals, 11 cages, where 6 specific animals cannot go into 4 specific small cages. The 6 large animals can use only the 7 large cages.</p><p><strong>Step 2:</strong> Place the 6 large animals into 7 available large cages. This is a permutation problem: P(7,6) = 7!/(7-6)! = 7!/1! = 5040 ways.</p><p><strong>Step 3:</strong> After placing 6 large animals, we have 5 remaining cages left (the 4 small cages + 1 remaining large cage) for the 5 remaining animals.</p><p><strong>Step 4:</strong> Arrange the 5 remaining animals in these 5 remaining cages: 5! = 120 ways.</p><p><strong>Step 5:</strong> By multiplication principle, total arrangements = 5040 × 120 = 604800.</p><p>∴ Answer: <strong>604800</strong></p>
Correct Answer: 604800