Final answer:
To determine the number of double-scoop ice cream options available with 15 flavors and three serving options, we calculate 15 choices for the first scoop times 15 for the second, resulting in 225 flavor combinations. Multiplied by three serving options, there are a total of 675 different double-scoop ice cream options.
Step-by-step explanation:
The student's question involves determining the number of different double-scoop ice cream options that could be made with 15 flavors and three serving options. To solve this, we will employ combinatorics, a mathematical concept that deals with counting combinations.
For each serving option (regular cone, waffle cone, or bowl), we can select two scoops of ice cream. Since the question states that the flavors can be repeated and the order matters, this is a permutations problem with repetition allowed. For each scoop, we have 15 choices of flavors, and since we want two scoops, we need to calculate 15 choices for the first scoop multiplied by 15 choices for the second scoop, giving us 15 x 15 = 225 combinations per serving option.
However, since we have three serving options, we need to multiply the number of combinations by three. Thus, the total number of different double-scoop options is 225 x 3 = 675.
This example illustrates the vast possibilities when it comes to customizing an ice cream order, similar to the endless combinations that can be created when selecting different types of pizza toppings.