Math Problem Statement
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Solution
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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.
Would you like further details or clarification on this?
Here are five related questions for deeper understanding:
- How do factorials work, and why are they used here?
- Why does the problem use 4 groups and what does each group represent?
- How would the answer change if the groups didn't need to be seated consecutively?
- What is the significance of the different factorial values (2!, 3!, etc.) for each group?
- 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.
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Math Problem Analysis
Mathematical Concepts
Factorials
Permutations
Combinatorics
Formulas
n! = n × (n-1) × ... × 1
Permutations Formula P(n, r) = n! / (n - r)!
Theorems
Fundamental Counting Principle
Factorial Growth
Suitable Grade Level
Grades 9-12
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