How many people are needed to guarantee that at least 4 people have a birthday in the same month?
How many people are needed to guarantee that at least 4 people have a birthday in the same month?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Question:**
How many people are needed to guarantee that at least 4 people have a birthday in the same month?
**Explanation:**
This question relates to the principles of the pigeonhole problem in probability and combinatorics. To ensure that at least 4 people share the same birthday month, we consider the worst-case scenario:
There are 12 months in a year. If we distribute people across these months and want to guarantee that at least one month will have at least 4 people, we use the formula:
\[ n = 12 \times (k-1) + 1 \]
where \( k \) is the number of people we want in the same month (in this case, 4). Therefore:
\[ n = 12 \times (4-1) + 1 \]
\[ n = 12 \times 3 + 1 = 37 \]
So, 37 people are needed to guarantee that at least 4 people have a birthday in the same month.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34cc5c6b-5450-4c98-93f3-d37eba173724%2F9c15e3bf-c221-40d5-bc9d-dc658cb5f983%2Fre5eulq_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
How many people are needed to guarantee that at least 4 people have a birthday in the same month?
**Explanation:**
This question relates to the principles of the pigeonhole problem in probability and combinatorics. To ensure that at least 4 people share the same birthday month, we consider the worst-case scenario:
There are 12 months in a year. If we distribute people across these months and want to guarantee that at least one month will have at least 4 people, we use the formula:
\[ n = 12 \times (k-1) + 1 \]
where \( k \) is the number of people we want in the same month (in this case, 4). Therefore:
\[ n = 12 \times (4-1) + 1 \]
\[ n = 12 \times 3 + 1 = 37 \]
So, 37 people are needed to guarantee that at least 4 people have a birthday in the same month.
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