For an art exhibit, Craig has to choose 3 ceramic mugs out of the 7 that he made over the summer. In how many ways can he arrange these 3 mugs in a row?

I'm not looking for the answer, but only an explanation for why this is a permutation problem and not a combination one.

because the order or arrangement matters. Otherwise, it'd just be a combination.

This is a permutation problem because the order of the selected ceramic mugs matters. In other words, the arrangement of the mugs in a row will be different depending on the order in which they are placed.

If it were a combination problem, the order of the selected mugs would not matter, and for this scenario, it would mean that all 3 selected mugs are considered as a set, without any regard to their specific positions in the row. But in this case, the order is important as an art exhibit usually involves displaying objects in a specific arrangement or sequence.

To understand why this problem involves permutations, let's first clarify the difference between permutations and combinations.

Permutations are arrangements of objects in a specific order, while combinations are selections of objects without considering their order.

In this case, Craig needs to choose 3 ceramic mugs out of the 7 he made and arrange them in a row. The order in which the mugs are arranged matters, as the mugs will be displayed in a particular order in the art exhibit. This makes it a permutation problem.

To further illustrate this, consider the following scenario:

Suppose Craig chose 3 mugs (Mug A, Mug B, and Mug C) out of the 7. Here are a few examples of different arrangements of these 3 mugs:

- Arrangement 1: Mug A, Mug B, Mug C
- Arrangement 2: Mug A, Mug C, Mug B
- Arrangement 3: Mug B, Mug A, Mug C
- Arrangement 4: Mug B, Mug C, Mug A
- Arrangement 5: Mug C, Mug A, Mug B
- Arrangement 6: Mug C, Mug B, Mug A

Each of these arrangements represents a different display that Craig could create with the 3 chosen mugs. Since the order matters, there are different possibilities for arranging the mugs in a row.

Therefore, the problem can be solved using permutations rather than combinations. To find the number of ways Craig can arrange the 3 mugs, you can use the formula for permutations:

nPr = n! / (n - r)!

Where n represents the total number of objects (in this case, 7 mugs), and r represents the number of objects chosen (in this case, 3 mugs).