# Are there 3000 types of sandwiches?

1. Jan 13, 2012

### Intervenient

1. The problem statement, all variables and given/known data
A delicatessen advertises that if offers 3000 types of sandwiches. If at this deli it is possible to have any combination of salami, turkey, bologna, corned beef, and ham with or without Swiss and/or American cheese on French, white or whole wheat bread, and possible additions of lettuce, tomato, and mayo, is the deli's advertisement true? Assume that a sandwich necessarily has bread and at least one type of meat or cheese.

2. Relevant equations

3. The attempt at a solution
I got 3072 as my solution, so the deli is being truthful.
My reasoning was that for the main topping there are 2^7 choices. I then multiplied that by 3, because there are 3 types of breads, but you can only have one of them. Then 2^3 different types of condiments.

However, knowing that the sandwich has to have at least one meat or cheese, that would make the choices down to 2^6 I think. Not sure :/

2. Jan 13, 2012

### I like Serena

Hi Intervenient!

If I understand your problem statement correctly, you need at least 1 choice from the first 7 main toppings.
Or put otherwise, it is not allowed to have none of the first 7 main toppings.

Si there are 2^7 combinations of which exactly 1 is not allowed.

3. Jan 13, 2012

### Staff: Mentor

okay so I got a different #
(2^7 -1) for meat / cheese selections (-1 it says at least one meat of cheese)