Answer to October 11, 2004 Problem |
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| The Subway Advertisement Problem |
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An advertisement on TV last summer claims that you can dine five nights a week
at Subway for the rest of your life and not have the same sandwich
twice. How many different sandwiches are possible? The Subway website lists seventeen different subs, which can be ordered in one of three ways: (1) as a six-inch sub, (2) as a twelve-inch sub, or (3) as a wrap. There are three additional wraps listed which can only be ordered as wraps. We will consider each of these to be a different sandwich (a six-inch meatball sub is different from a twelve-inch meatball sub which is different from a meatball wrap). For the subs, you have a choice of six different types of bread (wheat, white, etc.). Once you have selected your basic sandwich, then there are thirteen different toppings from which you may choose (cheese, pickles, onions, etc.), but you may select any number of these (or none of them). Finally, there are nine sauces from which to choose (mustard, mayonaise, vinegar, etc.). Most people probably just choose one of these items, so let's suppose that only one of these is chosen (or you may choose to have none of them).
Click here for the Subway menu I saw a similar ad on a Metro Train in Washington, DC this past summer. It read, "To see everything the Smithsonian has to offer, you'd have to view 4,348 items every day until you were ninety. Luckily, we've extended our hours this summer." While eating at Bintliff's American Cafe Restaurant in Portland, Maine, which is famous for its breakfasts, the menu stated that you can choose from over 25,600 different omelettes and fritatas. You are allowed to choose 4 items from the following: 10 different Meats, 19 different vegetables, and 10 different cheeses. |
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Solution to the Problem:
There are 18,350,080 different sandwiches. You could dine 5 nights a week for
70,577 years and not eat the same sandwich twice!
The total number of different sandwiches can be computed by multiplying the
number of basic sandwiches times the number of toppings times the number of sauces.
To compute the number of basic sandwiches, compute the number of subs and the number of
wraps separately and then add them together. The number of subs = 17 x 2 x 6 = 204
(17 is the number of subs listed in the menu, 2 is for six-inch or twelve-inch, 6 is
the number of bread choices). The number of wraps is 17 + 3 = 20 (17 is the number of six-inch subs
that can be ordered as wraps, 3 is for the three additional wraps listed). So, the total
number of basic sandwiches is 204 + 20 = 224.
To figure the number of ways that the toppings can be selected, you must consider every possible combination
of toppings and add them together. You must add together the following:
The number of choices for sauces is ten (any one of the nine sauces or none of them).
Therefore, the total number of different sandwiches is 224 x 8,192 x 10 = 18,350,080.
I contacted Subway about their advertisement but they did not respond.
After re-reading the problem, I realized that it is not clear that wraps can only be ordered one way. You don't get a choice for the six breads, so I counted it correct for those who figured the answer to be 26,542,080. Confession time: Actually, when I initially worked out the problem, I used only 11 toppings for some reason. It was only when I noticed three of you with the same answer that I decided to recheck my work! |
| 1. Larry Schwartz | Trumbull, Connecticut |
| 2. David & Judy Dixon | Bennettsille, South Carolina |
| 3. Jeffrey Gaither | Winchester, Virginia |
| 4. James Alarie | University of Michigan -- Flint, Flint, Michigan |
| 5. Dave Smith | Toledo, Ohio |
| 6. Misty Carlisle | Winchester, Virginia |
| 7. Arin Smith | Winchester, Virginia |