We here at SaberSmart are huge fans of buildyourown food restaurants like Chipotle, Moe’s, or Subway. The options appear endless, but they must come to an end somewhere. Not only does this give us a chance to introduce the fascinating topics of combinations and permutations, but it also allows us to update a Business Insider article from 2013 that went pretty viral as well! Back in their 2005 Annual Report, Chipotle’s CEO, Steve Ells, made the following claim: “Because customers can choose from four different meats, two types of beans and a variety of extras such as salsas, guacamole, cheese and lettuce, there’s enough variety to extend our menu to provide more than 65,000 choices”. Now, we know to be skeptical of numbers without context. However, we are skeptical of 65,000 because it actually seems too small! Results from combinations and permutations can add up fast, which is why our friend Walter Hickey updated this number back in 2013. Hickey came up with Chipotle having 655,360 ways to order a meal! However, this number has gotten outdated as well. Today, we plan on updating this for 2017. 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. Now it is time to gather our data from the Chipotle website.
For each category, except for Containers, we want to calculate the total number of combinations of choosing 0 through all of the possibilities. For instance, for the Rice category, we need to sum the combinations of choosing 0, 1, or 2 rices. Using the equation above, this turns out to be 1 + 2 + 1 = 4. After we calculate all of these sums, we simply multiply them together and then multiply that by the number of containers. This takes into account every combination from an empty burrito all the way to a soft corn taco stuffed with every ingredient.
Our final equation is then: 6 * 128 * 4 * 4 * 512 = 6,291,456 different meals! This number is almost ten times more than what Hendricks calculated back in 2013. Simply by adding one more type of taco and three more meat/filling options, the number of combinations skyrocketed. When viewed within the context of a single burrito or bowl, this is over 1 million options per container! To put it another way, you could eat a different burrito (only burrito) every day from Chipotle for over 2,870 years before having to consume a duplicate combination. At least you would pass this guy for the record number of consecutive days eating at Chipotle! As always, our code can be found on Github. The SaberSmart Team
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