A store is selling ice cream and has 100 different toppings available. A person may have none, some, or all toppings. How many different choices are there?

Question

for example chocolate sauce, cherries, and mint chips would be one choice. or butterscotch sauce and heath bar crumbles would be another choice.

Answer 1

Each topping is either included or it's not.  

 

So there are 2 choices for the 1st topping (it's either included or not), 2 choices for the 2nd topping, 2 choices for the 3rd topping, and so on for all 100 toppings.  

 

So the total number of choices is 2*2*2*... (100 times) which is 

 

2¹⁰⁰  

 

which is about 1.27 x 10³⁰ 

Answer 2

are you sure it's 100 different toppings ?

none would be 1 choice

one would give you 100 choices

two would give you: 100!/2!98!=100*99/2=4950 more choices

three would give you: 100!/3!97!=(100*99*98)/(3*2)=161,700 more choices

the number of choices should peak at 50 and then repeat themselves after that until you come to 100 toppings which is 1 choice

if you want to keep doing this, have fun !!!

maybe there is some sort of formula for all the choices from 0 to 100 that will make this simpler !?!?

I thought about the problem and finally realized the simple solution which Edward C. already has done for you.

This was my reasoning:

If there is 1 topping A, there are 2 choices, 0 or A; 2¹

If there are 2 toppings, A and B, the  choices are 0, A, B, and AB for a total of 4 choices; 2²=4

If there are 3 toppings, A, B, and C, you have the following choices:

0, A, B, C, AB, AC, BC, and ABC for a total of 8 choices; 2³=8

If there are 4 toppings there are 16 choices; figure them out-2⁴=16

There is a pattern, as you can see, so you can determine the answer for any number of toppings.