Question

At her favorite ice cream store, saga can put up to two toppings on an ice cream cone. if there are 106 different ways she can choose toppings (including the choice of no toppings), how many different toppings are there?

Answer 1

There are 106 different ways Saga can put 2 toppings on her ice cream, including the choice of no toppings. So Saga has 105 different ways of choosing the 2 toppings.

Let the total number of toppings be x. Saga has to choose 2 toppings, so this is a problem of combinations. Combination of 2 objects from can be written as nC2. There are total 105 ways to choosing the toppings, so we can write:

$nC2 = 105 \\ \\ \frac{n!}{2!*(n-2)!} =105 \\ \\ \frac{n(n-1)(n-2)!}{2*(n-2)!}=105 \\ \\ n(n-1)=105*2 \\ \\ n(n-1)=210 \\ \\ n=15$

Solving the quadratic equation above we get n=15.

This means 15 different toppings were there at the ice cream store.