Permutations & Combinations
Counting
Grade None
Question:
<p>In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, the maximum population of the city is</p>
<p>(1) \(2^{32}\)</p>
<p>(2) \((32)^2 - 1\)</p>
<p>(3) \(2^{32} - 1\)</p>
<p>(4) \(2^{32-1}\)</p>
Step-by-Step Solution
Key Concept: Each person's tooth configuration corresponds to a unique subset of 32 possible tooth positions (each position either has a tooth or doesn't). Since no person is toothless and each has ≤32 teeth, we count all non-empty subsets of a 32-element set.
<p><strong>Step 1:</strong> Model the problem: Consider 32 possible tooth positions in the mouth. Each position either contains a tooth or doesn't.</p><p><strong>Step 2:</strong> Each unique tooth configuration is a subset of these 32 positions. The total number of subsets of a 32-element set is 2^32.</p><p><strong>Step 3:</strong> Apply the constraint "no person without a tooth": We must exclude the empty set (which represents 0 teeth). This gives us 2^32 - 1 valid configurations.</p><p><strong>Step 4:</strong> Verify: Each subset represents a distinct tooth pattern. Since no two persons have identical tooth sets, the maximum population equals the number of distinct non-empty subsets.</p><p><strong>∴ Answer: 2^32 - 1</strong></p><p><em>Note: This equals 4,294,967,295 persons approximately.</em></p>
Correct Answer: C