You have the options of a plain, waffle or chocolate cone, you can have 1 scoop, 2 scoops, or 3 scoops. There are 36 flavour options. And you pick 2 toppings out of 10 different options. How many different Ice cream possibilities are there?
3 cone choices
3 scoop choices
36 flavour choices
10C2 topping choices
Total possibilities: 3*3*36*10C2 = ?
To determine the number of different ice cream possibilities, you need to multiply the number of options for each category: cone, scoops, and toppings.
For cones:
You are given three options: plain, waffle, or chocolate. Therefore, there are 3 possible choices for the cone.
For scoops:
You have three choices for the number of scoops: 1 scoop, 2 scoops, or 3 scoops. Therefore, there are 3 possible choices for the number of scoops.
For flavors:
You are given 36 flavor options. Since the number of scoops wasn't specified, let's assume you can have different flavors for each scoop. Therefore, for 1 scoop, you have 36 options. For 2 scoops, you have 36 options for the first scoop and another 36 options for the second scoop (36*36). For 3 scoops, you have 36 options for each scoop, resulting in a total of 36*36*36 possibilities.
For toppings:
You can choose 2 toppings from 10 options. To calculate this, you need to calculate the number of combinations for choosing 2 out of 10, which is denoted as "10 choose 2" or written as C(10, 2). This can be calculated using the formula: C(n, k) = n! / [k! * (n - k)!], where n is the total number of items and k is the number of items to be chosen.
Plugging in the values: C(10, 2) = 10! / [2! * (10 - 2)!] = 10! / [2! * 8!] = (10 * 9) / (2 * 1) = 45.
Therefore, there are 45 different combinations of toppings.
Now, to find the total number of ice cream possibilities, you multiply the number of options for each category: 3(cones) * 3(scoops) * (36 + 36*36 + 36*36*36)(flavors) * 45(toppings) = 3 * 3 * (36 + 1296 + 46656) * 45 = 3 * 3 * 48048 * 45 = 3,264,480.
Therefore, there are a total of 3,264,480 different ice cream possibilities.