There are 100 people. Is it true that at least 15 of them were born on the same day of the week?

To determine if at least 15 out of 100 people were born on the same day of the week, we can use the pigeonhole principle. 

The pigeonhole principle states that if \( n \) items are put into \( m \) containers, with \( n > m \), then at least one container must contain more than one item. 

Here, the "items" are the 100 people, and the "containers" are the 7 days of the week.

Let's apply the principle:

1. **Number of people (items):** 100
2. **Number of days of the week (containers):** 7

We want to find out if at least 15 people were born on the same day. First, let's distribute the 100 people as evenly as possible across the 7 days of the week. 

If each day has \( k \) people, then the total number of people is \( 7k \). We need to solve for \( k \) when the total is 100 people:

\[ 7k \approx 100 \]
\[ k = \frac{100}{7} \approx 14.2857 \]

Since \( k \) must be an integer, we consider \( k = 14 \) initially. This distribution gives each day 14 people:

\[ 7 \times 14 = 98 \]

Thus, 98 people can be distributed such that each day has 14 people. The remaining 2 people will then be distributed to some of the days, which means two of the days will have 15 people:

\[ 14 + 1 = 15 \]

Therefore, by the pigeonhole principle, since we can't perfectly distribute 100 people evenly among 7 days without exceeding 14 people per day, at least two of those days must have at least 15 people.

Hence, it is true that at least 15 of the 100 people were born on the same day of the week.

Would you like further details or have any questions about this explanation?

Here are 8 related questions you might find interesting:

1. How does the pigeonhole principle apply in other real-world scenarios?
2. What is the general formula for determining the minimum number of items per container in the pigeonhole principle?
3. How can we extend this principle to more complex problems?
4. What are some common misconceptions about the pigeonhole principle?
5. How is the pigeonhole principle used in probability theory?
6. Can the pigeonhole principle be applied to continuous data sets?
7. What are some famous problems solved using the pigeonhole principle?
8. How does the pigeonhole principle relate to combinatorics?

**Tip:** The pigeonhole principle can often be used to prove the existence of certain conditions without actually constructing examples. It’s a powerful tool in combinatorics and probability.

Explore how the pigeonhole principle applies to determining if at least 15 out of 100 people share the same birthday. Understand the distribution of people across days of the week and the mathematical certainty behind this principle. Suitable for high school students and anyone interested in combinatorics and probability theory.

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