One of the things that makes bagels so great are that they are highly customizable. There are seemingly infinite combinations that each person can customize to their individual tastes and preferences.
But when we say “infinite”, we’re using it as a placeholder for “really, really, really large number.” Let’s see if we can calculate how many bagel combinations are actually out there.
Methodology
Luckily, this is not the first time someone has tried to solve this type of problem. For inspiration, we looked to Counting Chipotle Combinations in which the author calculates that there are 6,291,456 possible meals you can order at Chipotle (using the menu as of 2017). This was their methodology (buckle up for a little math lesson):
First, we need to decide if we will be using combinations or permutations. The difference is simple. Permutations take into account the order, while combinations do not. The best example we have found explaining this discrepancy is with the Olympics. Say we have 8 athletes in an event. How many ways can we give out a gold, silver, and bronze medal? Well, 8 athletes could get gold, 7 could get silver, and 6 could get bronze. This permutation results in 8 * 7 * 6, or 336 ways.
On the other hand, say we have 3 gold medals. How many ways could we hand them out? Well, 8 athletes could get the first gold, 7 could get the next gold, and 6 could get the third gold. This looks like the same permutation above! That is true, however we end up duplicates in this list. Combinations therefore, are the permutations divided by the number of redundancies. Since we have 3 gold medals to give away, there are 3! or 6 variations and redundancies for every choice we pick. The total combination then is 336/6 which equals 56 combinations.
Remember, combinations sound simpler than permutations, because they are. There will always be fewer combinations than permutations.
Since order does not matter for our Chipotle meal, we are going to use the formula for combinations for our ingredients:
where n is the total number of ingredients in a category and k is how many ingredients we want to choose from that same strata.
By calculating the number of ingredients for Container, Filling, Rice, Beans, and Toppings, and number of of ingredients allowed per category, the final equation results in: 6 * 128 * 4 * 4 * 512 = 6,291,456 different meals!
Great! We’ve now got a template to work with – let’s apply this same approach to bagels.
Categories and ingredients
First step is to determine the categories, and “ingredients” per category.
To keep it simple, we’ll keep the categories as: Bagel, Spreads, Toppings, and Preparation
The tricky part is determining how many ingredients per category. Unfortunately, this list isn’t as straightforward as Chipotle’s simple menu. Thus, we compiled a list of “standard” bagels, spreads and toppings offered by a few NYC shops and a couple national bagel chains. This is what we found:
| Asiago Cheese |
| Blueberry |
| Cheddar Cheese |
| Chocolate Chip |
| Cinnamon Raisin |
| Cinnamon Sugar |
| Cranberry |
| Egg |
| Egg Everything |
| Everything |
| French Toast |
| Garlic |
| Gluten free everything |
| Gluten free plain |
| Jalapeño |
| Jalapeño Cheddar |
| Marble Rye |
| Oat Bran |
| Onion |
| Plain |
| Poppy Seed |
| Pumpernickel |
| Pumpernickel Everything |
| Rainbow |
| Salt |
| Sesame |
| Spinach |
| Whole Wheat |
| Whole Wheat Everything |
| Whole Wheat Sesame |
| Butter |
| Non-dairy butter |
| Jelly |
| Cinnamon Sugar |
| Peanut Butter |
| Almond Butter |
| Nutella |
| Honey |
| Hummus |
| Cream Cheese |
| Bacon Cheddar Cream Cheese |
| Bacon Maple Cream Cheese |
| Bacon Scallion Cream Cheese |
| Birthday Cake Cream Cheese |
| Blueberry Cream Cheese |
| Chipotle Bacon Cream Cheese |
| Chipotle Cream Cheese |
| Chive Cream Cheese |
| Cinnamon Walnut Raisin Cream Cheese |
| Cucumber Dill Cream Cheese |
| Guacamole Cream Cheese |
| Horseradish Dill Cream Cheese |
| Jalapeno Cheddar Cream Cheese |
| Jalapeno Cilantro Cream Cheese |
| Low-Fat Cream Cheese |
| Low-Fat Scallion Cream Cheese |
| Low-Fat Vegetable Cream Cheese |
| Lox Cream Cheese |
| Mixed Berry Cream Cheese |
| Nutella Cream Cheese |
| Olive Cream Cheese |
| Oreo Cream Cheese |
| Plain Cream Cheese |
| Salsa Cream Cheese |
| Scallion Cream Cheese |
| Scallion Tofu Cream Cheese |
| Strawberry Cream Cheese |
| Spinach Artichoke Cream Cheese |
| Sun-Dried Tomato Cream Cheese |
| Tofu Cream Cheese |
| Vegetable Cream Cheese |
| Vegetable Tofu Cream Cheese |
| Whipped Cream Cheese |
| Za’atar Cream Cheese |
| “Salads” |
| Egg salad |
| Chicken salad |
| Whitefish salad |
| Tuna salad |
| Salmon salad |
| Eggs |
| Cheese |
| Lox |
| Bacon |
| Sausage |
| Ham |
| Avocado |
| Tomato |
| Onion |
| Capers |
| Lettuce |
| Spinach |
| Jalapeno |
| Toasted |
| Not-toasted |
the calculation
Let’s start simple and see how many combinations there are if you only consider our 30 types of bagels with just one spread:
- 0 or 1 spreads
- 0 toppings
- Toasted or non-toasted
The result is: 30 * 49 * 1 * 2 = 2,940 combinations
Let’s see what happens if you allow for 1 additional spread:
- 0, 1 or 2 spreads
- 0 toppings
- Toasted or non-toasted
The possible combinations increases dramatically: 30 * 1177 * 1 * 2 = 70,620 combinations
Now let’s add toppings to the mix. Let’s try adding just 1 topping:
- 0, 1 or 2 spreads
- 0 or 1 toppings
- Toasted or non-toasted
The result is: 30 * 1177 * 14 * 2 = 988,680 combinations
Whoa, we’re almost at 1 million combinations! But who wants just one topping? Let’s allow up to 5:
- 0, 1 or 2 spreads
- 0, 1, 2, 3, 4 or 5 toppings
- Toasted or non-toasted
The result is: 30 * 1177 * 2380 * 2 = 168,075,600 combinations
Conclusion
Using a non-exhaustive list of spreads, toppings, and preparation methods, we estimate that there are over 168 million possible bagel combinations. This means you could eat a different bagel every day for 460,274 years without eating the same combination twice.
Now, we must note that our ingredients list is a gross over-simplification of reality. Practically speaking, there are many more bagels, spreads, toppings, and preparation methodologies (e.g. scooped, open-faced, etc.). And of course, there is no actual limit on the number of spreads or toppings one can use.
Even within the toppings category, there is an incredible amount of variation we’re not considering: e.g. dozens of different ways to prepare the eggs, the number of eggs that go on the bagel, individual types of cheeses, non-traditional meats or veggies, condiments, etc.
Not to mention another category we ignored that can be an incredibly important variable: which shop the bagel comes from.
Taking those into account, the possible number of combinations quickly skyrockets into unpronounceable numbers. So for all intents and purposes – yes, the number of ways a bagel can be customized is infinite.
So what?
Why bother going through the effort of this calculation? It ties directly to our central thesis about why bagels are so great:
Savory or sweet, toasted or not toasted, scooped, flagel, on its own or as a sandwich…with dozens of flavors of bagels, cream cheeses, and toppings to choose from – bagels allow each and every person to have a truly distinct favorite.
We love nothing more than hearing about which of these infinite combinations is your favorite. Which of the infinite possibilities is your favorite bagel combination? Please comment below!
