Question

A delicatessen advertises that it offers over 30,000 varieties of sandwiches. If at this deli it is possible to have any combination of salami, turkey, bologna, corned beef, roast beef, and ham with or without Swiss, Provolone, and/or American cheese on French, white, whole wheat, or multigrain bread, and possible additions of lettuce, tomato, mustard, and mayonnaise, is the deli's advertisement true? Assume that a sandwich necessarily has bread and at least one type of meat or cheese.

Answer 1

There are 9 types of meat/cheese. These are listed below:

1. Salami
2. Turkey
3. Bologna
4. Corned beef
5. Roast beef
6. Ham
7. Swiss cheese
8. Provolone cheese
9. American cheese

These 9 items has two choices - i) can be added. ii) cannot be added

Hence, the meat/cheese can be chosen in ways.
However, one of these ways include adding of nothing, and this shall be removed.
Hence, all possible ways of adding at least a meat/cheese is

There are 4 types of bread. These are listed below:

1. French bread
2. White bread
3. Whole wheat bread
4. Multigrain bread

A burger will contain any one type of bread. The bread can be chosen in ways.

There are 4 possible additions. These are listed below:

1. Lettuce
2. Tomato
3. Mustard
4. Mayonnaise

These 4 items has two choices - i) can be added. ii) cannot be added
Hence the additions can be chosen in ways.
Note: A customer may choose none of the additions. Hence, we do not need to substract the no-addition case.

Hence, total number of combinations possible are:




It is known that,


Hence, theoritically, delicatessen's advertisement that it offers over 30,000 varieties of sandwiches is true.

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