Question

Five students \((A, B, C, D,\) and \(E)\) are scheduled to present papers in class. (a) How many ways can this be arranged? (b) How many ways can this be arranged without \(B\) speaking before \(A\) ? (c) How many ways can this be arranged if \(A\) speaks immediately before \(B\) ?

Short answer

(a) 120 ways, (b) 60 ways, (c) 48 ways.

Step-by-step solution

Understand the Problem

We need to determine the number of different ways the presentations can be arranged for parts (a), (b), and (c). This involves permutations and some constraints.

Calculate Total Arrangements

For part (a), calculate the total number of arrangements of the five students using permutations. Since there are no restrictions, the total number of ways to arrange the students is given by the factorial of the number of students: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Arrangements with B not before A

Part (b) requires counting the number of arrangements where B does not speak before A. To find this, first calculate the total number of arrangements: 120. Then note that for every arrangement where B speaks before A, there is a corresponding arrangement where A speaks before B. Therefore, half of the arrangements will meet this criterion. \[ \frac{5!}{2} = \frac{120}{2} = 60 \]

Arrangements with A immediately before B

For part (c), we want to place A and B together, treating them as a single unit or block. There are 4 blocks to arrange: (AB), C, D, and E. Calculate the permutations of these 4 blocks:\[ 4! = 24 \]For each arrangement of these blocks, A and B can be ordered within the block as AB or BA. So, multiply by 2:\[ 4! \times 2 = 24 \times 2 = 48 \]

Conclusion

To summarize, the solutions to the parts of the question are: (a) 120, (b) 60, and (c) 48.

Key concepts

Factorial

Factorials are a mathematical concept that helps calculate the number of ways to arrange a set of objects. In the context of permutations, factorials provide an efficient way to compute all possible orders or sequences.When dealing with permutations, the factorial of a number, denoted as \( n! \), represents the product of all positive integers up to \( n \). For example, the factorial of 5, or \( 5! \), is calculated as:\[ 5 \times 4 \times 3 \times 2 \times 1 = 120 \]This formula helps to quickly determine how many different ways a set of items can be arranged when no constraints are applied, as seen in the arrangement problem of five students. Remembering how factorials work can be key in solving various combinatorial problems, providing a foundation for understanding more complex constraints and arrangements.

Constraints

Constraints in permutations limit the number of valid arrangements of a set of items based on specific conditions. For example, in our arrangement problem, one constraint is that student B should not speak before student A. By understanding constraints, we can determine how many arrangements meet specific requirements. Consider these points: - Constraints affect the symmetry of arrangements. For every sequence where one condition is met, there is likely a counterpart where it isn't. In our problem, this symmetry means half the arrangements don't have B before A. - Constraints often lead to creative thinking. By identifying and understanding them, you can find shortcuts, like splitting permutations into manageable parts or blocks, as seen when A and B are treated as a single unit in another part of the problem. Embracing constraints helps efficiently narrow down permutations and discover valid solutions.

Arrangement Problem

Arrangement problems involve finding the number of possible orders for a set of items, accounting for any constraints. These problems are common in situations like scheduling, distributing tasks, or solving puzzles where order matters. When tackling arrangement problems, consider: - Identifying all items to arrange. Our example involved five students, which simplifies to calculating arrangements of these individuals. - Noting any constraints or requirements that might alter the straightforward approach. This could be as simple as an order constraint---like B not speaking before A---or as complex as requiring certain items to appear together, as seen with A immediately before B. In the final part of the exercise, students A and B form a block to ensure their order, reducing the problem to four items. Therefore, understanding both general and specific arrangement problems can equip you with essential strategies for countless scenarios. Arrangement problems use core mathematical concepts like permutations and factorials, allowing for systematic solutions even with added constraints.