On a flight with 138 passengers, each passenger had the choice of peanuts or pretzels for a snack and the coice of chicken or beef for dinner. The beverage service offered 12 beverages. A flight attendant later remarked that no 2 passengers had the same request. Is this possible if each passenger took one snack, one dinner, and one beverage? Is this possible if each passenger took no more than 1 selection from each category? Is this possible if the flight attendant, being a new recruit, offered the snack by asking, "peanuts, pretzels, or both?" and each passenger took no more than one dinner or beverage?

Question

On a flight with 138 passengers, each passenger had the choice of peanuts or pretzels for a snack and the coice of chicken or beef for dinner. The beverage service offered 12 beverages. A flight attendant later remarked that no 2 passengers had the same request. Is this possible if each passenger took one snack, one dinner, and one beverage? Is this possible if each passenger took no more than 1 selection from each category? Is this possible if the flight attendant, being a new recruit, offered the snack by asking, "peanuts, pretzels, or both?" and each passenger took no more than one dinner or beverage?

Answer 1

To determine if it is possible for no two passengers to have the same request, we can consider each scenario separately:

1. Each passenger takes one snack, one dinner, and one beverage:
Since there are only 2 snack options (peanuts or pretzels), 2 dinner options (chicken or beef), and 12 beverage options, it is not possible for no two passengers to have the same request if they had to pick one item from each category. This is because the number of passengers (138) is greater than the total number of unique combinations of snacks, dinners, and beverages (2 x 2 x 12 = 48).

2. Each passenger takes no more than 1 selection from each category:
In this case, if the passengers are allowed to choose either peanuts or pretzels, it is possible for no two passengers to have the same request. Each passenger can simply choose a different option from the available selections. Similarly, if they can choose either chicken or beef, and one of the 12 beverages, it is still possible for no two passengers to have the same request.

3. The flight attendant offers the snack options as "peanuts, pretzels, or both," and each passenger takes no more than one dinner or beverage:
If the passengers can choose either peanuts, pretzels, or both, it is possible for no two passengers to have the same request. Each passenger can choose a combination that is different from the others. However, since each passenger can only choose one dinner option and one beverage option, it is not possible for no two passengers to have the same request in these categories. This is because the number of passengers (138) exceeds the total number of unique combinations of dinners (2) and beverages (12) (2 x 12 = 24).

In conclusion, it is possible for no two passengers to have the same request if each passenger takes no more than one selection from each category and the snack options are "peanuts, pretzels, or both."