Deducing the Minimum Number of People with All Four Items: A Case Study

Imagine a survey conducted on 100 people where various characteristics are noted. The data reveals that 72 had a wallet, 85 wore brown shoes, 58 carried a briefcase, and 98 wore a ring. The question at hand is: how many people must have had at least all four items? This article will explore this intriguing problem using the principle of inclusion-exclusion and set theory.

Understanding the Data and Constraints

The total population being surveyed is 100 individuals. Let's break down the numbers:

Number of people with a wallet (A): 72 Number of people with brown shoes (B): 85 Number of people with a briefcase (C): 58 Number of people with a ring (D): 98

The sum of people having each item is:

72 85 58 98 313

Since there are only 100 people surveyed, we need to find the overlap to determine the minimum number of people who must have all four items.

Solving Using the Principle of Inclusion-Exclusion

To find the minimum number of people who must have all four items, we can use the principle of inclusion-exclusion. The formula is:

Minimum with all four items A B C D - 3 * Total people surveyed

Substituting the given values:

Minimum with all four items  313 - 3 * 100  313 - 300  13

This result indicates that at least 13 people must have had a wallet, wore brown shoes, carried a briefcase, and wore a ring.

A Decoupled Approach to Verify the Result

Another method to confirm the result involves considering the worst-case scenario for each item. Here’s a step-by-step breakdown:

People with a briefcase: 58 Removing people without a ring from this group: 58 - 2 56 Removing people without a wallet from this group: 56 - 28 28 Removing people without brown shoes from this group: 28 - 15 13

This method also confirms that at least 13 people must have all four items.

Verifying the Calculations

Alternatively, we can calculate the worst possible scenario where people are missing the items:

People not having a wallet: 100 - 72 28 People not wearing brown shoes: 100 - 85 15 People not carrying a briefcase: 100 - 58 42 People not wearing a ring: 100 - 98 2

Now, let's combine these figures to find the number of people not having any of these items:

People not having all the items  100 - 28 - 15 - 42 - 2  13

This calculation confirms that at least 13 people must have all four items.

Conclusion

In conclusion, using both the principle of inclusion-exclusion and a worst-case scenario analysis, we have demonstrated that the minimum number of people who must have all four items (a wallet, brown shoes, a briefcase, and a ring) is 13. This problem highlights the importance of set theory and the principle of inclusion-exclusion in solving real-world problems involving overlapping sets.