<p>Find the number of ways in which 16 constables can be assigned to patrol 8 villages, 2 for each.</p>
Step-by-Step Solution
Key Concept: This is a partition problem where we must divide 16 identical constables into 8 distinct groups of 2 each, then account for the overcounting due to identical group arrangements. The formula is 16!/(2!^8 × 8!) since the villages are distinguishable but the order within each pair and the order of selection doesn't matter.
<p><strong>Step 1: Recognize the structure</strong></p><p>We need to assign 16 constables to 8 distinct villages with exactly 2 constables per village. Each pair of constables is indistinguishable in terms of their role.</p><p><strong>Step 2: Select 2 constables for village 1</strong></p><p>Number of ways = C(16,2)</p><p><strong>Step 3: Select 2 constables for village 2 from remaining 14</strong></p><p>Number of ways = C(14,2)</p><p><strong>Step 4: Continue for all 8 villages</strong></p><p>Total = C(16,2) × C(14,2) × C(12,2) × C(10,2) × C(8,2) × C(6,2) × C(4,2) × C(2,2)</p><p><strong>Step 5: Simplify using the formula</strong></p><p>This equals: 16!/(2!×2!×2!×2!×2!×2!×2!×2!) = 16!/(2!^8)</p><p><strong>Step 6: Calculate</strong></p><p>16!/(2!^8) = 20922789888000/256 = <strong>81729648000</strong></p><p>Or expressed as: 16!/(2^8) = <strong>16!/256</strong></p><p><strong>∴ Answer: 16!/2^8 = 81,729,648,000 or equivalently 20922789888000/256</strong></p>
Correct Answer: 16