We will make it more routine to discuss about some of the Daily Magic Spell problems that the majority of students who have attempted the question on given day did not get it correct. We are interested in hearing your thoughts and interpretation about the problem and offer feedback to your approach in determining the solution. We encourage everyone to participate in the discussion! Feel free to post your questions if you have any!
Without further ado, we will begin with the question given on January 11, 2017.
Suppose that a restaurant sells $7$ different burgers and $4$ different salads. Two people decide to order different things, but both order a burger or both order a salad. How many different ways can this happen?
To approach this problem, note that two people must each order a burger, or a salad. That is, if Person $A$ orders a burger, Person $B$ orders a burger. If Person $A$ orders a salad, Person $B$ must order a salad.
In the case when Person $A$ orders a burger, Person $A$ has $7$ options to select from. After Person $A$ made his choice, there are $6$ remaining burgers for Person $B$ to select from. By the Product Rule, this yields $7 \times 6 = 42$ ways that two people can order different burgers.
In the case when Person $A$ orders a salad, Person $A$ has $4$ options to select from. After Person $A$ made his choice, there are $3$ remaining salads for Person $B$ to select from. Again, by the Produce Rule, this yields $4 \times 3 = 12$ ways that two people can order different salads.
By the Sum Rule, this implies that there is a total of $42 + 12 = 54$ ways to do this.
I believe that the most common mistake with the question is the interpretation of the problem. This is very common in counting problems. One possible misinterpretation is the phrase "two people decide to order different things." This phrase may indicate that the $7$ different burgers and $4$ different salads are free to choose from and therefore each person could potentially order a burger and salad. This is incorrect since either both people must order a burger or both people must order a salad.
Do you also have some thoughts about the problem? How did you misinterpret the problem? Let us know below!