Students are awarded 5 grades A, B, C, D, E how many students must be there so that atleast 5 students get the same grade ?

To solve the problem of determining how many students must be there so that at least 5 students get the same grade, we can use the **Pigeonhole Principle**.

### Explanation:
1. **Pigeonhole Principle**: If \( n \) items are put into \( m \) containers, with \( n > m \), then at least one container must contain more than one item.

2. In this case:
   - The "items" are the students.
   - The "containers" are the grades (A, B, C, D, E).

   There are 5 grades, so we have 5 containers.

3. We want to ensure that at least 5 students get the same grade.

### Step-by-Step Solution:
- If we have fewer than 5 students with the same grade, it’s possible that all students are distributed among the grades without having any grade with at least 5 students. This distribution will happen as follows:
  - \( 4 \) students in Grade A.
  - \( 4 \) students in Grade B.
  - \( 4 \) students in Grade C.
  - \( 4 \) students in Grade D.
  - \( 4 \) students in Grade E.
  
  This distribution totals \( 4 \times 5 = 20 \) students, and each grade has fewer than 5 students.

- Now, if we add one more student, bringing the total to 21 students, by the Pigeonhole Principle, this additional student must go into one of the grades, thereby giving that grade at least 5 students.

### Conclusion:
Therefore, **21 students** are required to ensure that at least 5 students receive the same grade.

Would you like more details or have any questions about this solution? Here are some related questions:

1. How can the Pigeonhole Principle be applied to different scenarios?
2. What is the minimum number of students required so that at least 4 students receive the same grade?
3. How does the Pigeonhole Principle apply in real-life situations?
4. What if there were 6 different grades instead of 5?
5. How does the problem change if we want at least 6 students to get the same grade?
6. Can you explain the general formula for applying the Pigeonhole Principle?
7. What is a real-world example where the Pigeonhole Principle is used in computer science?
8. How does the concept of combinatorics relate to this problem?

**Tip:** The Pigeonhole Principle is a useful tool in combinatorics and probability, helping to determine minimum numbers in situations involving distribution or allocation.

Discover how to determine the minimum number of students required so that at least 5 students receive the same grade using the Pigeonhole Principle. Explore a detailed explanation and step-by-step solution to this combinatorial problem.

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