The other day my student brought me a homework problem from First-Year Stats at U of Winnipeg. Here is the problem:
1b). An ice cream shop offers an end of season special Fall
deal for a fixed cost, where you can choose two of twenty flavors, two of eight
toppings, and one of four cones. How many different ice cream treat
combinations are available?
In the preamble to the problem sheet, the prof asks the students to show what formula they are using, but she doesn't ask them to state any assumptions about what is being sought. Where do you begin with a problem like this? You are getting two scoops of ice cream, two toppings, and one cone. What makes this kind of problem challenging is that you have to decide if you care what order the choices are made. Since the prof doesn't tell you whether or not you should care, you have to decide based on the real-life interpretation of the problem.
Is it a different order if you get chocolate with peppermint sprinkles and butterscotch with chocolate sauce, versus chocolate with chocolate sauce and butterscotch with peppermint sprinkles? You tell me. Is it different if you have black cherry on the bottom and cappucino on the top, or the other way around? Or would both of those count as the same order? Your guess is as good as mine.
Whether or not you choose to count those permutations as different or the same makes all the difference in the world as to how you would do the calculation. But the professor doesn't want you to think about that. She wants you to pick a formula and show your work, which means show how you plug the numbers into the formula. That's what passes for "education" in today's univeristies.