Questions: The Wycoffs are planning their next family night. They always have dinner out somewhere and then do something fun together. There are 2 adults and 4 children in the family. Each family member is allowed 3 meal suggestions, and each child is allowed 4 activity suggestions. Assuming no family members choose the same thing, how many different family night possibilities are there?

Solution Steps

To determine the number of different family night possibilities, we need to calculate the total number of meal suggestions and activity suggestions. Each family member has 3 meal suggestions, and each child has 4 activity suggestions. We then multiply the total number of meal suggestions by the total number of activity suggestions to get the total number of possibilities.

Each family member (2 adults + 4 children = 6 members) can suggest 3 meals. Therefore, the total number of meal suggestions is given by:

\[ \text{Total Meal Suggestions} = (2 + 4) \times 3 = 6 \times 3 = 18 \]

Each child (4 children) can suggest 4 activities. Thus, the total number of activity suggestions is:

\[ \text{Total Activity Suggestions} = 4 \times 4 = 16 \]

The total number of different family night possibilities is the product of the total meal suggestions and the total activity suggestions:

\[ \text{Total Possibilities} = \text{Total Meal Suggestions} \times \text{Total Activity Suggestions} = 18 \times 16 = 288 \]

Final Answer