Problem 115 - A Complete Resource Book in Mathematics for JEE Main

Question

In a city no person has 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, then the maximum population of the city is (A) \(2^{32}\) (B) \(2^{32}-1\) (C) \(2^{32}+1\) (D) None of these

Step-by-Step Solution

Answer

The maximum population is \(2^{32} - 1\).

Step 1: Understanding the Problem

We are considering people who can have between 1 and 32 teeth. For each position, a tooth can either be present or not, leading to different combinations of teeth placements for each person.

Step 2: Calculating Possible Combinations for a Given Number of Teeth

For any given set of positions (with a maximum of 32 positions), there are 2 possible choices (tooth present or not) for each, leading to a potential of \(2^{32}\) different combinations.

Step 3: Eliminating the Case of Zero Teeth

Given that no person can have no teeth, we subtract 1 from the \(2^{32}\) combinations to account for the combination where no teeth are present.

Step 4: Summarizing the Maximum Population

Thus, the number of potential unique arrangements of teeth positioning for the city's population is \(2^{32} - 1\). This excludes the empty set that represents no teeth.

Key Concepts

Binary Combinations

Binary combinations simplify complex arrangements by reducing choices to two options: tooth present or not in this context. For each tooth position, a person either has a tooth or they don't. This creates two outcomes per position which leads to exponential growth in possibilities.

For 32 positions (representing potential tooth locations), the total number of possible combinations is calculated using the binary formula: \[2^{32}\]This accounts for every possible configuration of teeth, including scenarios where no teeth are present. Such binary enumeration forms the foundation of many combinatorial problems.

Population Calculation

Population calculation in this problem requires counting unique tooth arrangements while respecting given constraints. Each permutation of teeth represents a unique individual. With every position offering two possibilities, the city's potential population equals the number of binary combinations.

• The total unique combinations start at \(2^{32}\).
• To respect the rule that no person has zero teeth, we exclude one combination where no tooth is present.
• Thus, the population becomes \(2^{32} - 1\).

This exclusion is essential to ensure realism in our population count.

Positional Arrangement

Positional arrangement is key to understanding how small variations lead to large numbers of unique outcomes. Each of the 32 positions for a tooth offers a distinct opportunity to create a new arrangement.

• Every person differs by at least one tooth's presence or absence.
• This difference in dental arrangement encodes identity within the city.
• A full set of 32 allows us to explore the maximum range of variations in tooth setup.

While only a slight shift, such arrangements result in significantly different combinations, highlighting the power of disposition in combinatorics.

Set Theory

Set theory provides a structured method to examine the combinations of teeth. Here, each person's tooth set maps to a distinct binary string of length 32. These binary strings collectively form a mathematical set representing all possible populations.
Consider the set solution:

• The power set represents all subsets of these positions – this includes the empty set (no teeth).
• Removing the empty set gives us \(2^{32}-1\) valid combinations.
• This ensures compliance with the rule that every individual has at least one tooth.

Set theory thus concretizes concepts like identities and restrictions mathematically, providing clarity and precision to the task of enumeration.