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Ways of ordering freshman = 2!  
Ways of ordering sophomores = 3!  
Ways of ordering juniors = 3!  
Ways of ordering seniors = 5!  
Ways of ordering 4 groups = 4!

Ways to seat the students at the table if students of the same grade must be seated consecutively = 2! × 3! × 3! × 5! × 4!  
= 207,360

So, there are 207,360 ways to seat the students at the table if students of the same grade must be seated consecutively.

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Here are five related questions for deeper understanding:

1. How do factorials work, and why are they used here?
2. Why does the problem use 4 groups and what does each group represent?
3. How would the answer change if the groups didn't need to be seated consecutively?
4. What is the significance of the different factorial values (2!, 3!, etc.) for each group?
5. How do the different group sizes impact the overall seating arrangement?

**Tip:** Factorial functions grow very quickly with larger numbers, making them useful for counting arrangements and permutations.

Ways to seat the students at the table if students of the same grade must be seated consecutively. Factorials are used for different groupings of students based on grade.

Learn how to calculate the number of seating arrangements for students grouped by grade level using factorials. This problem uses combinatorics and factorial growth to determine seating for groups of freshmen, sophomores, juniors, and seniors. Suitable for high school students in Grades 9-12.

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