4. Probability
Counting
2:23 minutesProblem 3.T.7b
Textbook Question
7. There are 16 students giving final presentations in your history course.
b. Presentation subjects are based on the units of the course. Unit B is covered by three students, Unit C is covered by five students, and Units A and D are each covered by four students. How many presentation orders are possible when presentations on
the same unit are indistinguishable from each other?
Verified step by step guidance1
Step 1: Understand the problem. The total number of students is 16, and they are grouped by units: Unit A (4 students), Unit B (3 students), Unit C (5 students), and Unit D (4 students). Presentations on the same unit are indistinguishable, meaning the order within each unit does not matter.
Step 2: Recall the formula for permutations of indistinguishable items. When arranging a set of items where some are indistinguishable, the total number of arrangements is given by the formula: \( \frac{n!}{k_1! \cdot k_2! \cdot \dots \cdot k_m!} \), where \( n \) is the total number of items, and \( k_1, k_2, \dots, k_m \) are the counts of indistinguishable items in each group.
Step 3: Apply the formula. Here, \( n = 16 \) (total students), and the groups are \( k_1 = 4 \) (Unit A), \( k_2 = 3 \) (Unit B), \( k_3 = 5 \) (Unit C), and \( k_4 = 4 \) (Unit D). Substitute these values into the formula: \( \frac{16!}{4! \cdot 3! \cdot 5! \cdot 4!} \).
Step 4: Simplify the factorials. Compute the factorials for \( 16! \), \( 4! \), \( 3! \), and \( 5! \), but do not calculate the final result yet. The factorial of a number \( n! \) is the product of all positive integers from 1 to \( n \). For example, \( 4! = 4 \cdot 3 \cdot 2 \cdot 1 \).
Step 5: Divide \( 16! \) by the product of \( 4! \cdot 3! \cdot 5! \cdot 4! \). This will give the total number of possible presentation orders where presentations on the same unit are indistinguishable. Leave the result in its simplified form if needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Permutations and Combinations
Permutations and combinations are fundamental concepts in combinatorics that deal with counting arrangements and selections. Permutations refer to the different ways to arrange a set of items where order matters, while combinations refer to selections where order does not matter. In this question, since presentations on the same unit are indistinguishable, combinations are more relevant for calculating the total arrangements.
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Multinomial Coefficient
The multinomial coefficient is a generalization of the binomial coefficient and is used to count the ways to divide a set of items into multiple groups. It is expressed as n!/(k1! * k2! * ... * kr!), where n is the total number of items, and k1, k2, ..., kr are the sizes of the groups. In this scenario, it helps determine the number of distinct arrangements of students presenting on different units.
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Indistinguishable Objects
Indistinguishable objects refer to items that cannot be differentiated from one another within a set. In this problem, the students presenting on the same unit are indistinguishable, meaning their order does not affect the overall arrangement. This concept is crucial for simplifying the calculation of possible presentation orders, as it reduces the total number of unique arrangements.
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