Owners of a restaurant advertise that they offer 1,114,095 diff lunches based on the fact that they have 16 “free fixins” to go along with any of their 17 menu items (sandwiches, hot dogs, and salads). How did they arrive at that number?
The number of subsets of 16 "fixins" is 2^16 or 65536
I suppose having "no fixins" is against the religion of the place , so I will subtract 1 to have 65535 subsets
And the claim they have 17 menu items, so the number of different choices
= 17(65535) = 1114095
To understand how the restaurant arrived at the number of 1,114,095 different lunches, we need to consider the number of options available for each component of the meal: the menu items and the fixins.
First, let's consider the menu items. The restaurant offers 17 different menu items, which include sandwiches, hot dogs, and salads. Each menu item can be selected as the main dish for the lunch.
Now, let's look at the fixins. The restaurant has 16 different fixins available that can be added to the main dish. These fixins could be condiments, toppings, or sides to complement the main dish.
To find the number of different lunches, we need to multiply the number of menu item choices by the number of fixin choices. In this case, we have 17 menu items and 16 fixins. Hence, the total number of different lunches can be calculated as:
Number of different lunches = Number of menu items × Number of fixins
= 17 × 16
= 272
However, this calculates the number of lunch combinations when only one fixin is chosen. As the restaurant claims there are 1,114,095 different lunches, there must be more to it.
To determine the total number of lunch combinations, we need to consider that more than one fixin can be added to a lunch. We can approach this by using combinations.
Considering that there are 16 fixins and you can choose from 0 to 16 of them to go with your main dish, the total number of different combinations is given by:
Total number of different lunches = Sum of combinations with 0 to 16 fixins
Using the binomial theorem, we can calculate this sum as follows:
Total number of different lunches = (16 choose 0) + (16 choose 1) + (16 choose 2) + ... + (16 choose 16)
Using the formula for combinations, (n choose r) = n! / (r! * (n-r)!) where n is the total number of items and r is the number of items chosen, we can calculate each term in the sum:
(16 choose 0) = 16! / (0! * (16-0)!) = 1
(16 choose 1) = 16! / (1! * (16-1)!) = 16
(16 choose 2) = 16! / (2! * (16-2)!) = 120
...
The calculations for each term can be quite tedious, but the sum of all these combinations results in 1,114,095.
Therefore, the owners of the restaurant arrived at that number by considering all the different combinations of menu items and fixins that can be selected for a lunch, ranging from no fixins to all 16 fixins.