We are working on permutations and combinations and the problem reads as follows: 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?
Can you have more than one item of the non-cheese items?
Can you have more than one kind of cheese?
e.g.
could you have a "loaded" sandwich with all 5 items and all the cheeses ?
If you want just one item and one cheese, the answere is easy.
that would be 5x5 or 25 ways.
BUT, could you have a sandwich with just one kind of cheese and nothing else?
Can you see how one must be very clear when phrasing a question like this?
Oh yes I understand very well how important it is to be very clear when asking a question like this, but it's one of the questions on an assignment with multiple choice answers. The best that I could come up with looks like 2x2x2x2x2x5=160....(5 for the various cheeses, 2 for with lettuce or without, 2 for with tomatoes or without, etc.) which is one of the optional answers....but when we ask the instructor to clarify she just tells us to look at the equation and plug in the numbers....I was just hoping my answer would be justified when compared to another's answer.
To find the number of different cheese sandwiches possible, we need to consider the number of choices for each ingredient (mayonnaise, lettuce, tomatoes, pickles, sprouts) and the types of cheese available.
Since there are no restrictions on the number or combination of ingredients, we can assume that each ingredient can be present or absent in a sandwich.
Let's break down the problem step by step:
1. First, we consider the choices for each ingredient:
- Mayonnaise: It can be either present or absent (2 choices).
- Lettuce: It can be either present or absent (2 choices).
- Tomatoes: They can be either present or absent (2 choices).
- Pickles: They can be either present or absent (2 choices).
- Sprouts: They can be either present or absent (2 choices).
2. Next, we consider the types of cheese available:
- There are 5 types of cheese available.
By multiplying the number of choices for each ingredient and the number of cheese types, we can find the total number of different cheese sandwiches possible:
2 (mayonnaise) x 2 (lettuce) x 2 (tomatoes) x 2 (pickles) x 2 (sprouts) x 5 (cheese types) = 160 different cheese sandwiches possible.
So, there are 160 different cheese sandwiches possible.