Vincenzo makes a sandwich using four kinds of Italian lunch meat: Genoa, prosciutto, Calabrese and capicola and two kinds of Italian cheese: provolone and asiago. Vincenzo'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 Vincenzo arrange the meat and cheese on his sandwich?

Question

Vincenzo makes a sandwich using four kinds of Italian lunch meat: Genoa, prosciutto, Calabrese and capicola and two kinds of Italian cheese: provolone and asiago. Vincenzo'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 Vincenzo arrange the meat and cheese on his sandwich?

Answer 1

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Answer 2

To find the number of different ways Vincenzo can arrange the meat and cheese on his sandwich, we can use the concept of permutations.

First, let's determine the available options for placing the layers of meat. Vincenzo has four types of Italian lunch meat: Genoa, prosciutto, Calabrese, and capicola. Since each type of meat can only be used once, there are 4 options for the first layer, 3 options for the second layer, 2 options for the third layer, and 1 option for the fourth layer. This gives us a total of 4! (read as "4 factorial") ways to arrange the meat layers.

Similarly, for the cheese layers, Vincenzo has two types of Italian cheese: provolone and asiago. To ensure that the two types of cheese are not next to each other, we need to consider the following possibilities:

Case 1: Provolone is placed in an odd-numbered layer (1st or 3rd) and asiago is placed in an even-numbered layer (2nd or 4th).
In this case, Vincenzo has 2 options for the first cheese layer (provolone or asiago) and 2 options for the second cheese layer (the remaining type of cheese). Therefore, there are 2 × 2 = 4 arrangements.

Case 2: Provolone is placed in an even-numbered layer (2nd or 4th) and asiago is placed in an odd-numbered layer (1st or 3rd).
Similarly, for this case, Vincenzo has 2 options for the first cheese layer and 2 options for the second cheese layer. So, there are 2 × 2 = 4 arrangements.

Now, to find the total number of arrangements, we can add the number of arrangements for each case: 4 + 4 = 8.

Therefore, Vincenzo can arrange the meat and cheese on his sandwich in 4! (24) ways, given that the two types of cheese are not next to each other.