Final answer:
To find the number of different ways this can happen, we use the principle of inclusion-exclusion. After counting the total number of possibilities without any restrictions, we count the number of possibilities where at least one person gets both an ice-cream cone and a chocolate-chip cookie. Subtracting the latter from the former gives us the final result.
Step-by-step explanation:
To find the number of different ways this can happen, we can use the principle of inclusion-exclusion. Let's first count the total number of possibilities without any restrictions. Each person can either get an ice-cream cone or a chocolate-chip cookie, so there are 2 choices for each person. Since there are 8 people, there are a total of 2^8 = 256 possibilities.
Now, let's count the number of possibilities where at least one person gets both an ice-cream cone and a chocolate-chip cookie. We can think of this as a separate event where one person gets both items and the rest of the people choose between the remaining items.
The person who gets both items can be any one of the 8 people. The remaining 7 people can each choose between 2 items for a total of 2^7 = 128 possibilities. Therefore, the number of possibilities where at least one person gets both items is 8 * 128 = 1024.
Finally, to find the number of different ways this can happen, we subtract the number of possibilities where at least one person gets both items from the total number of possibilities:
Total number of possibilities - Number of possibilities with at least one person getting both items = 256 - 1024 = -768
However, the result is negative, which means there are no possible ways for this scenario to happen. This suggests that there might be a mistake or inconsistency in the question. Please double-check the question or provide more information if possible.