using the pigeon hole principal, how many students would i need to ensure that 5 of them get the same score from a range of 0 to 100

The Pigeonhole Principle states that if \( n \) items are placed into \( k \) containers, and if \( n > k \), then at least one container must contain more than one item.

In your case:

- The "items" are the students.
- The "containers" are the possible scores, which range from 0 to 100. That gives you 101 possible scores (since 0 to 100 includes 101 integers).

You want to know how many students you need to ensure that at least 5 of them have the same score. In this context, we want to place students (items) into score ranges (containers), ensuring that at least one score contains 5 students.

### Solution:
- There are 101 possible scores (0 to 100), so there are 101 containers.
- To avoid having 5 students with the same score, the maximum number of students you can have is \( 4 \times 101 = 404 \), where each score is assigned to at most 4 students.

If you add just one more student (making the total 405), by the pigeonhole principle, at least one score must have 5 students.

Thus, **you would need 405 students** to ensure that at least 5 of them have the same score.

Would you like further details or have any questions?

Here are some related questions:
1. How would the answer change if the score range was 0 to 50 instead of 0 to 100?
2. Can the Pigeonhole Principle be applied to a continuous range of scores?
3. What if you wanted 6 students to have the same score? How would the number change?
4. How does the Pigeonhole Principle apply in probability theory?
5. Can you think of a real-world example where the Pigeonhole Principle might be applied?

**Tip:** The Pigeonhole Principle can be surprisingly useful for proving the existence of solutions in many mathematical problems where distribution is involved.

Using the pigeon hole principle, how many students would I need to ensure that 5 of them get the same score from a range of 0 to 100?

Discover how the Pigeonhole Principle can be applied to find the minimum number of students needed to ensure that 5 students have the same score in a range from 0 to 100. This detailed explanation includes core combinatorics concepts and is suitable for high school students (Grades 9-12).

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