Question

There are t different caps \(c_{1}, c_{2}, c_{3}, c_{4}\) and \(c_{5}\) and 5 different \(b_{0 x_{4}}\) \(B_{1}, B_{2}, B_{3}, B_{4}\) and \(B_{5}\). The capacity of each box is sufficlent ob accomodate all the 5 caps. If \(B_{1}\) and \(B_{5}\) have the caps \(C_{1}\) and \(C_{5}\) among themselves, in how many ways can you arrange the caps among the 5 boxes? (a) 400 (b) 485 (c) 500 (d) 365

Short answer

Answer: There are 12 ways to arrange the caps among the 5 boxes.

Step-by-step solution

First arrange \(C_{1}\) and \(C_{5}\) in boxes \(B_{1}\) and \(B_{5}\)

We have two caps \(C_{1}\) and \(C_{5}\) and two designated boxes \(B_{1}\) and \(B_{5}\) for them. Suppose \(C_{1}\) is placed in \(B_{1}\) and \(C_{5}\) is in \(B_{5}\). There is another case where \(C_{5}\) is placed in \(B_{1}\) and \(C_{1}\) in \(B_{5}\). So, we have 2 ways to arrange \(C_{1}\) and \(C_{5}\) among \(B_{1}\) and \(B_{5}\).

Place the remaining caps in the rest of the boxes

Now we have 3 caps left (\(C_{2}, C_{3},\) and \(C_{4}\)) and 5 boxes (\(B_{1}, B_{2}, B_{3}, B_{4},\) and \(B_{5}\)). We have to place these 3 caps in the remaining empty spaces of the boxes such that each box contains one cap. We will remove the cases when \(B_{1}\) and \(B_{5}\) have 2 caps in them as that would violate the given condition. To count the number of ways to distribute the remaining caps: Choose a box for \(C_{2}\) from the 3 available boxes (\(B_{2}, B_{3},\) and \(B_{4}\)): \(3\) ways Choose a box for \(C_{3}\) from the remaining 2 available boxes: \(2\) ways Place \(C_{4}\) in the last remaining box: \(1\) way. This gives \(3 \times 2 \times 1 = 6\) ways to arrange the remaining caps in the remaining boxes.

Combine the arrangements

From Step 1, we know that there are 2 ways to arrange \(C_{1}\) and \(C_{5}\) in boxes \(B_{1}\) and \(B_{5}\). And from Step 2, we know that there are 6 ways to arrange the remaining caps in the other boxes. To get the total number of ways, we just multiply the number of ways from Step 1 and Step 2. Total number of ways = 2 (ways to arrange \(C_{1}\) and \(C_{5}\)) * 6 (ways to arrange the remaining caps) = 12 So there are 12 ways to arrange the caps among the 5 boxes, none of the given options is correct.

Key concepts

Combinatorial Arrangements

In combinatorics, arrangements refer to the different ways of organizing objects. When dealing with distinguishable objects, such as different caps and boxes, the number of arrangements becomes an important aspect.
This exercise provides a practical example of arranging caps among designated boxes. Although the original choices on the list did not include twelve, calculating the number of ways to organize objects in these specific restrictions is a core concept.

• Designated arrangements: When certain objects are restricted to specific positions or groups, focusing on these arrangements is crucial.
• Total configurations: This involves calculating all possible organizations, keeping in mind any limitations provided.

Here, the desired task is to place caps in certain boxes with particular restrictions, helping us comprehend how to approach and solve shuffling problems in combinatorics.

Counting Principle

The fundamental counting principle is a key concept used to determine the number of possible combinations or arrangements in a problem. Simply stated, it tells us if one event can occur in 'm' ways and another, independent event can occur in 'n' ways, then together the events can occur in m*n ways.
In the context of the given task, the counting principle aids us in finding a solution by breaking down the problem into simpler deciding steps:

• Firstly, arranging two caps in their specific boxes ( C_{1} and C_{5} ), resulting in 2 possible ways.
• Secondly, organizing the remaining three caps in the leftover boxes, yielding 6 ways, calculated as 3 x 2 x 1, considering the choices decrease as boxes are filled.

Thus, by using the counting principle, we can find the total number of ways by simply multiplying the results of these independent events to arrive at 12 distinct arrangements.

Permutation Problems

Permutation problems are a branch of combinatorics where the main task is arranging objects in specific sequences or orders. When it comes to permutation and combination, permutation primarily focuses on the sequence and placement within a given set.
Given the challenge of arranging 5 different caps among 5 boxes with specific constraints, we are engaged in solving a permutation problem. This is due to the fact that each cap must be placed in its unique box in a specific order.
The initial arrangement of caps C_{1} and C_{5} enforces a sequence, while placing the remaining caps highlights the permutation aspect of arranging.

• Prioritizing sequence: It’s essential to determine a sequence for specific items when solving permutation tasks.
• Incorporating restrictions: Constraints, such as allocated positions or mandatory pair placements, shape how combinations are calculated.

Understanding permutation allows us to untangle the intricate patterns present in such arrangements.