Four items from three different departments of Metro Department Store will be featured in a one-page newspaper advertisement as shown in the following diagram.
If items from the same department must be in the same row, how many arrangements are possible?

A) 83,024
B) 82,944
C) 82,815
D) 83,116
E) 82,929

Question

Four items from three different departments of Metro Department Store will be featured in a one-page newspaper advertisement as shown in the following diagram. ​  ​
If items from the same department must be in the same row, how many arrangements are possible?
​
A) 83,024 
B) 82,944 
C) 82,815 
D) 83,116 
E) 82,929

Quizplus · Accepted Answer

To determine the number of arrangements, we can use the multiplication principle:

- For the first row, we have 4 choices of items from the first department. 
- For the second row, we have 2 choices of items from the second department (since one item has already been used). 
- For the third row, we have 2 choices of items from the third department (since one item has already been used). 
- For the fourth row, we have 3 choices of items from the second and third departments combined (since one item from each department has already been used).

Therefore, the total number of arrangements is:

4 x 2 x 2 x 3 = 96

However, we also need to consider the order of the items within each row. There are 4! = 24 ways to arrange the items within each row.

So the final answer is:

96 x 24 x 24 x 24 = 82,944.

Therefore, the best choice is B.

Four Items from Three Different Departments of Metro Department Store