Problem 4 - Algebra for JEE ADVANCED

Question

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 a. \(2^{32}\) b. \((32)^{2}-1\) c. \(2^{32}-1\) d. \(2^{32-1}\)

Step-by-Step Solution

Answer

Option c. \(2^{32} - 1\) is correct.

Step 1: Understanding the Problem

We need to determine the maximum population of a city based on a unique set of teeth for each person. Each position of a tooth can either have a tooth present or not, with no two people having the same set of positioning.

Step 2: Analyzing the Maximum Sets of Teeth

Each person can have up to 32 teeth. At each of the 32 possible positions, there can either be a tooth or not, leading to a total of possibilities being a power of 2.

Step 3: Calculating Total Possibilities

Since each tooth position can either be filled or empty, and there are 32 positions, the possible combinations are given by the formula for combinations: \[ 2^{32} \]

Step 4: Adjusting for No Identical Sets

Since the problem states no two persons have identical sets of teeth and no one is toothless, we must subtract 1 from the total, to exclude the combination where there are zero teeth.

Step 5: Final Calculation

Thus, the maximum population can be given by:\[ 2^{32} - 1 \]

Step 6: Deciding the Answer

From the given options, the answer that matches our calculation is: c. \(2^{32} - 1\).

Key Concepts

Binary Possibilities

When solving problems involving combinations like this one, a key concept is binary possibilities for each position or decision. Here, you have 32 positions in a set of teeth. Each position can have two options: either there is a tooth or there isn't. This type of binary choice is represented mathematically using powers of 2.

To make this clearer, think about flipping a coin. Each flip has two outcomes, heads or tails — it's a binary outcome. If you flip a coin once, you have 2 outcomes: heads and tails. Flip it twice, and you have 4 combinations: head-head, head-tail, tail-head, tail-tail, which can be calculated as \(2^2\). If you extend this process, flipping the coin 32 times, you have \(2^{32}\) possible outcomes. In the case of our exercise, it's the positions of teeth either having a tooth or not.

Thus, when you have 32 positions, each with 2 possibilities, the total number of combinations of teeth that could exist is \(2^{32}\). This gives you a solid foundation for understanding how many unique combinations are possible.

Population Estimation

Population estimation often involves determining the maximum possible number of individuals given specific conditions or constraints. In this case, we have constraints based on teeth configurations. To solve such problems, we use combinatorics to figure out the possible unique combinations available.

Here, we're using the formula \(2^{32} - 1\) to specify not only the possible configurations but also applying it's adjustment because nobody can have zero teeth. By subtracting one combination (zero teeth configuration), we obtain the true maximum population — hence arriving at \(2^{32} - 1\) configurations. This concludes that if each person must have at least one tooth, this becomes the upper bound estimate for the population.

Such precise population estimations are common in theoretical exercises, especially when you want to predict upper limits based on defined parameters or characteristics like the ones demonstrated here.

Exclusion Principle

The Exclusion Principle plays a crucial role when estimating combinations that fall under specific conditions; in this case, each person must have a non-empty set of teeth. To address this, we utilize the exclusion principle to adjust for specific non-allowed scenarios.

In combinatorics, this principle entails deliberately counting configurations and then systematically excluding certain categories — here, those with no teeth at all. Students may often miscalculate outcomes by neglecting the exclusion principle, as it ensures these configurations do not unintentionally inflate the total number of valid combinations. In other words, it's crucial when you exclude the configuration of having zero teeth to ensure the solutions comply with the exercise's conditions.

So, remember that the exclusion principle is what modifies the standard calculation from \(2^{32}\) to \(2^{32} - 1\), ensuring no one is mistakenly considered as toothless. It's an excellent tool for refining outcomes to meet problem specifics much like the present scenario.