A deli offers its cheese sandwich with various combinations of mayonnaise, lettuce, tomatoes,
pickles, and sprouts. 5 types of cheese are available. How many different cheese sandwiches are
possible?
the toppings on each sandwich can be done in 8 ways, the number of subsets of 3 elements. This will include the nullset, or no toppings.
The wording is not clear as to how many cheeses can go on a sanwich.
e.g. can you have a sandwich with 3 different cheeses?
I will assume we can only have one of the 5 cheeses, so the number of different sandwiches is 5(8) or 40.
If you can have any combination of cheese, there would be 2^5 - 1 or 31 ways to have just the cheese sandwich.
(I subtracted one to exclude the case of no cheeses. After all it is called a "cheese sandwich")
So in that case there would be 5(31) or 155 different sandwiches.
Your call.
To find the number of different cheese sandwiches that are possible, we need to consider the number of choices we have for each component of the sandwich.
Let's break it down:
1. Cheese: There are 5 types of cheese available. This means we have 5 options to choose from for the cheese.
2. Mayonnaise: We need to determine if it's optional or required. If it's optional, we have 2 choices: either we can have mayonnaise or not have mayonnaise. If it's required, then we only have 1 choice: to have mayonnaise.
3. Lettuce: Similar to mayonnaise, we need to determine if it's optional or required. If it's optional, we have 2 choices: to have lettuce or not have lettuce. If it's required, then we only have 1 choice: to have lettuce.
4. Tomatoes: Again, we need to determine if it's optional or required. If it's optional, we have 2 choices: to have tomatoes or not have tomatoes. If it's required, then we only have 1 choice: to have tomatoes.
5. Pickles: Once more, we need to determine if it's optional or required. If it's optional, we have 2 choices: to have pickles or not have pickles. If it's required, then we only have 1 choice: to have pickles.
6. Sprouts: Similarly, we need to determine if it's optional or required. If it's optional, we have 2 choices: to have sprouts or not have sprouts. If it's required, then we only have 1 choice: to have sprouts.
Now, we can multiply the choices for each component together to find the total number of different cheese sandwiches:
Total number of different cheese sandwiches = (Number of cheese choices) * (Number of mayonnaise choices) * (Number of lettuce choices) * (Number of tomato choices) * (Number of pickle choices) * (Number of sprout choices)
Total number of different cheese sandwiches = 5 * 2 * 2 * 2 * 2 * 2
Simplifying this expression, we have:
Total number of different cheese sandwiches = 5 * 2^5
Total number of different cheese sandwiches = 5 * 32
Total number of different cheese sandwiches = 160
Therefore, there are 160 different cheese sandwiches that are possible.