Problem 115 - A Complete Resource Book in Mathematics for JEE Main 2017

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: Understand the Problem Setup

Each person in this hypothetical city has a different pattern of teeth, ranging from 1 to 32 teeth. Since no one person can have zero teeth and no one can have more than 32, we must calculate all possible unique combinations of teeth distribution between 1 and 32 teeth.

Step 2: Determine the Teeth Combinations for Each Individual

For every tooth, there are two possibilities: either the tooth is present or it is absent. Therefore, for 32 possible tooth positions, there are \(2^{32}\) combinations of presence or absence of teeth.

Step 3: Account for No Toothless Individuals

Since no person can be toothless (i.e., have 0 teeth), we must subtract the one combination where all teeth are absent from our previous total. This leaves us with \(2^{32} - 1\) possible combinations representing individuals with at least one tooth.

Step 4: Conclusion

The maximum population of the city, considering each individual having a unique set of teeth from those combinations, is \(2^{32} - 1\). This accounts for the restriction that no person has zero teeth.

Key Concepts

Binary Combinations in Combinatorics

Binary combinations play a crucial role in solving problems related to choices that can be represented in a binary manner, meaning each choice can be either "yes" or "no," "1" or "0," "present" or "absent." In the context of this problem, binary combinations help determine the various possible configurations of tooth presence for up to 32 teeth, and these configurations are used to estimate the maximum number of unique individuals in a city.

When it comes to teeth, each position can either have a tooth or not, which directly translates into binary decisions. Hence, with 32 teeth, the problem essentially involves calculating all the possible combinations of 32 binaries.

• The binary nature means that for each tooth, you have two options: it is either present (1) or absent (0).
• The number of total binary combinations for 32 dentures is found using the formula for binary combinations, which is expressed as \( 2^{32} \).

This formula calculates the maximum number of different combinations possible without considering additional constraints. Understanding binary combinations simplifies complex real-world problems by translating them into binary yes-or-no questions, which can greatly aid in fields such as computer science, mathematics, and genetics.

Understanding Unique Combinations

Unique combinations are combinations where each one is distinctly different from the others. This concept is vital in understanding the makeup of any collection where each item or person's identification must be unique, like in this city's population scenario.

In the exercise, each person must have a unique combination of teeth arrangements. This is because no two people can have an identical pattern of teeth, emphasizing the importance of unique combinations.

• While \( 2^{32} \) accounts for all possible combinations of teeth presence and absence, not all these combinations are valid.
• There is one combination where no teeth are present, and it must be excluded since no person is toothless, leading to \( 2^{32} - 1 \) unique and valid combinations.

The goal is to ensure that the combinations account for every possibility as long as it adheres to the constraints given. Recognizing and counting unique combinations efficiently is instrumental in tasks such as database management, coding theory, and cryptography.

Population Estimation Using Combinatorics

Population estimation in combinatorics involves applying the concepts of combinations to estimate the number of different possible configurations or individuals in a set. This is vital in contexts where each configuration or individual must be guaranteed to be distinct.

In this problem, to estimate the maximum population of the city based on unique teeth combinations, we leveraged the combinatorial concept of binary choice for each of the 32 positions. However, a careful adjustment was needed for realistic applicability:

• Start from \( 2^{32} \) possible combinations by considering all variations provided by binary decisions at each tooth position.
• Subtract the one combination where no teeth are selected to meet the problem's constraints (i.e., everyone must have at least one tooth).

This results in \( 2^{32} - 1 \) possible variations, giving us the maximum population capacity under the given constraints. Understanding how to estimate populations using combinatorial principles is broadly useful in fields such as urban planning, epidemiology, and market analysis where predictive modeling of various scenarios is required.