How many combinations are possible

Question

I'm trying to figure out the number of possible combinations that can be created in a frozen yogurt shop. There are 18 different flavors of froyo and 80 toppings. customers can get 1 or more flavors of froyo and one or more toppings. The possibilities seem endless - but the I'm curious about how to figure out the actual number of combinations.

Answer 1

The number of ways to have frozen yogurt combinations is 1 up to 18 flavors! 
That 18C1 + 18C2 + 18C3 + ... + 18C17 + 18 C18 which (if you know the binomial theorem you'll understand this) equals exactly 2¹⁸ - 1.  Indeed a large number.

 

Now # of toppings, if any combination from NONE up to all 80 is possible, that equals exactly 2⁸⁰.

 

The product of these two numbers is the answer desired. Of course, it is not economically feasibble, so this is the theoretical maximum...very big, but a lot less than infinity, which I cannot begin to describe!