Subway Sandwiches

Exploration:
Question: How many different combinations are there for sandwiches in Subway?

Background: In Math, we recently learned (or have been reminded of) a simple way to count the different outcomes, and Subway is infamous for the wide variety of sandwiches it can make, especially with the "Make Your Own Meal." This is where you choose the type and size of bread, the sandwich you want, topics, sauces, sides, beverages, and other things. For the following exploration, I will be using the menu below, which comes from this website.

*Note: Since my blog broke, the website has changed. This is a relatively old version of the menu. Some of my math will not match this version of the menu.

Several things to note beforehand:

1. You have the choice at Subway to either toast or not toast the bread.
2. You don't need to have everything. You can have no extra sauce or toppings if you want.
3. I will assume you are here for a sandwich, maybe wanting a salad or side orders. You are thirsty in this scenario.
4. You can have up to 4 vegetables per foot-long sandwich, 2 per 6-inch.
1. I will assume you have at least 1 topping.
2. The order you order these in will matter. Lettuce then tomato will be considered different from the other way around.
3. You can have 1 sauce only, assuming you choose to.
5. You can have repeats of the vegetables, so 2 tomatoes and 2 lettuce are options.
6. All my information either comes directly from the Subway Website, or from past experience.

Process (All values are in bold):

The first thing we need to determine is: What kind of sandwich do we want? There are 16 types of sandwiches, as shown in the center of the image. After that, we determine the size. We can have either 6-inch or foot-long. This brings us to a total of 32 possibilities. Next, we can bring in the type of bread. There are 6 types of bread, and they can be toasted or not toasted. This brings us to 384 (32*12) possibilities so far. Now, we can have up to 4 toppings. Having at least 1, we'll set this to 8*9*9*9, so that you have at least 1 topping, and say no to the other 3. It's a bit strange, the way I'm doing it, as you can say no to a second, and end up with a 3rd and 4th, but this is the way I'm sticking with. Just for simplicity. Now, we're up to 2,239,488 possibilities already. Finally, we add a sauce. You can say no. According to the above image, there are 8 sauces, with a 9th being "none." With this, we are now up to 20,155,392 possibilities. How about an extra in the sandwich? The bottom left shows extra items you can add into the sandwich: Extra cheese, beef strips, double meat, or extra pepperoni. Because each of these are different options, we now multiply our previous number by 16, as it's "cheese or no cheese, meet or no meet, (...) ". We now have 322,486,272 possibilities. We can't forget about the salad you may or may not want. Multiply by 5 to get one of these or none: 1,612,431,360 possibilities. There are also 5 side orders you can choose to get, a 6th would be none, and I'm assuming you can only get one: 9,674,588,160. Last but not least, you're gonna need a drink to quench the thirst of all those different combinations: Multiply by 3, since we're assuming you have to get a drink. 29,023,764,480. This is the number of possibilities you can get in terms of sandwiches.

That would be annoying to make a tree diagram for... So many branches that keep going until you get to over 29 billion outcomes...

This large number means you would be able to go to Subway for over 29 billion days and get something different every day. This would be equivalent to 79,517,162.959 years.

My math may or may not be wrong, I would just like to point that out.