Assane N. answered • 06/30/24
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The question could be more clear, but let’s consider two scenarios.
First scenario:
Let's approach this step-by-step:
1. First, we need to understand what the question is asking. We're looking for the number of different orders in which the inspector can visit all the machines.
2. This is a problem of permutation. We're arranging the order of visiting all machines.
3. The key here is to recognize that we're not concerned with the order within each department, but rather the order of visiting the departments themselves.
4. So, we can consider each department as a single unit. We have 3 units (departments) to arrange.
5. The number of ways to arrange 3 distinct items is simply 3! (3 factorial).
6. 3! = 3 × 2 × 1 = 6
Therefore, there are 6 different ways for the inspector to route their inspection.
These 6 ways are:
1. Department 1 → Department 2 → Department 3
2. Department 1 → Department 3 → Department 2
3. Department 2 → Department 1 → Department 3
4. Department 2 → Department 3 → Department 1
5. Department 3 → Department 1 → Department 2
6. Department 3 → Department 2 → Department 1
This solution assumes that the inspector must visit all machines in one department before moving to the next department.
Second scenario:
if we do NOT consider each department as a single unit but rather have the inspector do the inspection on a machine basis, then
To determine the number of different ways the inspector can inspect the machines, we need to find the total number of ways to arrange the sequence of inspections.
The inspector has a total of:
- 3 machines in the first department
- 5 machines in the second department
- 2 machines in the third department
This makes a total of 3+5+2=10 machines.
The number of different ways the inspector can inspect all the machines is simply the number of permutations of these 10 machines, which can be calculated as 10! (10 factorial).
