Ariel makes a sandwich using four kinds of Italian lunch meat: A, B, C and D and two kinds of Italian cheese: X and Z. Ariel's sandwich has a single layer of each type of meat and a single layer of each kind of cheese, but he also wants to make sure that the two types of cheese are not next to each other. How many different ways can Ariel arrange the meat and cheese on his sandwich?

6!-5!*2=480

To find the number of different ways Ariel can arrange the meat and cheese on his sandwich, we can consider the two types of cheese (X and Z) separately and then multiply the results.

First, let's consider the cheese. Since Ariel wants to make sure that the two types of cheese are not next to each other, we can start by placing one type of cheese (let's say X) in a fixed position. This leaves us with three possible positions for the other type of cheese (Z), as it cannot be placed directly next to X.

Now, let's move on to the meat. Ariel has four kinds of Italian lunch meat (A, B, C, and D) to choose from, and he wants to have a single layer of each kind. We can arrange these four kinds of meat in 4! (4 factorial) ways.

Combining the results, there are 3 ways to arrange the types of cheese and 4! (24) ways to arrange the types of meat. Multiplying them together:

3 (ways to arrange cheese) x 4! (ways to arrange meat) = 3 x 24 = 72

Therefore, there are 72 different ways Ariel can arrange the meat and cheese on his sandwich while ensuring that the two types of cheese are not next to each other.