Final answer:
There are 630 possible orders for two scoops of different flavor ice cream given 21 flavors and a choice between a sugar cone, waffle cone, or cup, not considering the order of the flavors.
Step-by-step explanation:
The question asks how many possible orders there are if a person orders two scoops of different flavor ice cream from an ice cream shop that has 21 flavors and offers a choice between a sugar cone, waffle cone, or cup, assuming they don't care about the order of flavors. This is a combination problem where order does not matter and we need to select 2 different flavors out of 21. We use the combination formula which is given by C(n, k) = n! / (k! * (n - k)!), where n is the number of items to choose from, and k is how many items we are choosing.
Here, n is 21 (flavors) and k is 2 (scoops), so we calculate C(21, 2). This evaluates to 21! / (2! * (21 - 2)!) = 21! / (2! * 19!) = (21 * 20) / (2 * 1) = 210 different combinations of flavors. Since the order of scoops doesn't matter, we don't need to multiply by 2. Now, as there are 3 different types of cones or cups (sugar cone, waffle cone, or cup), we multiply the number of flavor combinations by 3. So, the total number of orders is 210 * 3 = 630 possible orders.