0

The poster in McDonald’s says you can make a happy meal by choosing a main, a side and a drink. These are the choices:

Main: Hamburger, Cheeseburger, Chicken Nuggets, Crispy Snack Wrap, Seared Snack Wrap

Side: Fries, Apple pieces

Drink: Soft drink, Thickshake, Juice, Water

How many different Happy Meals can we construct?

If we have a Happy Meal every week, is it possible to cover all the possible Happy Meals exactly once so that every week we have a different choice for all three parts than we did last week?

I'm trying to solve this permutation problem, my first approach for the first question is using factorial

$5! \times 2! \times 4! = 5760 $(ways)

Is this the correct way to do it? And for the second question, I don't really have any idea to solve it, can you guys help me out. All helps are appreciate

Righter
  • 811
Peter
  • 65
  • 2
    Do not haphazardly throw out factorials and binomial coefficients and such without understanding what they are used for and represent. This question is far more fundamental than questions that involve factorials and rely on more basic principles of counting, namely here the rule of product. First, learn the rule of product and understand it completely. Only then should you move on to using factorials who, by the way, can be explained using those more basic fundamentals. – JMoravitz Apr 28 '21 at 12:55

1 Answers1

1

Your answer to the first question is far too high: you have five main choices, two side choices and four drink choices, making $5 \times 2 \times 4$ overall possible combinations, since you want one of each.

On the second, the answer is yes. Note that if you want no matches week-to-week then you will have to alternate sides each week, but you have enough flexibility on mains and drinks to do this. It is not too difficult to make a list of $40$ combinations to prove this. Here is one possibility:

HFS CAT NFJ WAW SFS HAT CFJ NAW WFS SAT HFJ CAW NFS WAT SFJ HAW CFS NAT WFJ SAW HFT CAS NFW WAJ SFT HAS CFW NAJ WFT SAS HFW CAJ NFT WAS SFW HAJ CFT NAS WFW SAJ

Henry
  • 157,058