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Chapter 6: Problem 33

Show that there are at least six people in California (population: 39 million) with the same three initials who were born on the same day of the year (but not necessarily in the same year). Assume that everyone has three initials.

Short Answer

Expert verified
At least 6 people in California share the same three initials and birthday.

Step by step solution

01

Calculate the Total Possible Combinations of Initials

To find the total number of possible combinations of three initials, consider that each initial can be any of the 26 letters in the English alphabet. Therefore, there are 26 choices for each of the three initials.\[ 26 \times 26 \times 26 = 17,576 \]So, there are 17,576 possible combinations of three initials.
02

Calculate the Number of Possible Birthdays

There are 365 days in a common year. Hence, a person's birthday can be on any of these 365 days.
03

Calculate the Total Number of Unique Combinations

Each person can have any one of 17,576 three-initial combinations and be born on any one of 365 days. Thus, the total number of unique combinations of initials and birthdays is:\[ 17,576 \times 365 = 6,416,040 \]There are 6,416,040 unique combinations.
04

Apply the Generalized Pigeonhole Principle

California's population is 39 million people. Using the generalized pigeonhole principle, if there are more people than there are unique combinations (39 million > 6,416,040), then at least one unique combination must be shared by multiple people. To find how many people share at least one combination of initials and birthday, divide the population by the number of unique combinations:\[ 39,000,000 \text{ people} \big/ 6,416,040 \text{ combinations} \ \big\rfloor = 6.08 \]Thus, there must be at least 6 people sharing the same three initials and birthday combination.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics, a branch of mathematics, concerns the counting, arrangement, and combination of elements in sets. It is crucial in solving problems related to probability, statistics, and various fields of science and technology. Consider a simple example: the calculation of possible initials in this exercise. Each initial can be one of 26 letters. Therefore, the number of combinations of three initials is calculated as: \[ 26 \times 26 \times 26 = 17,576 \]
Initials Combinations
This problem considers initials, which are typically the first letters of a person's first, middle, and last names. The total number of initials combinations is fundamental in combinatorial problems. When calculating combinations of three initials, considering each initial can be any of 26 letters in the alphabet gives: \[ 26 \times 26 \times 26 \rightarrow 17,576 \]. Now, consider these initials combined with birth dates, which yields unique personal identifiers in our context.
Birthday Problem
The generalized pigeonhole principle helps us solve the birthday problem by demonstrating that if there are more people than there are unique combinations, repetitions are inevitable. Here, we combine birthdays with initials. There are 365 possible birthdays in a year. Thus, the total unique combinations of three-initial and birthdays are: \[ 17,576 \times 365 = 6,416,040 \]. Given California's population of 39 million, many people share the same initials and birthday, leading to the conclusion that at least six people will likely have the same three initials and the same birthday.

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