# 6 possible options on each of 3 days. How many arrangements are possible? (Is this as terribly ambiguous a problem as I think?)

Here's a textbook question from Essentials of Probability & Statistics, a very standard-fare undergrad intro to stats/probability text. This early chapter problem is posed as follows:

Suppose registrants at a large convention are offered 6 sightseeing tours on each of 3 days. In how many ways can an attendee arrange to go on a sightseeing tour planned by this convention?

Now, to my mind, this is an obviously ambiguous problem. There are immediately several possible ways to interpret this, none of which is any more rational to select than any other (since there is no additional information or constraint given), and all of which have wildly divergent answers. As far as I can tell, there are 4 possible straightforward interpretations:

1. 18 ways: The attendee chooses just one tour from all the 18 options available over the 3 days. This means they're attending only one tour during the entire convention, and they just pick one of 18.

2. 20 ways: The attendee chooses 3 distinct tours out of the 6 available over the 3 days (no repeats). This means they're attending one tour each day, but the order doesn't matter, so it's just $$\binom{6}{3} = 20$$.

3. 120 ways: The attendee chooses 3 distinct tours out of the 6 available, but they can arrange them in a different order to produce a new "arrangement." Then we have $$P(6,3) = 120$$.

4. 216 ways: The attendee has 6 choices of tours each day, and they can repeat tours on different days. This means they're attending one tour each day and could potentially attend the same tour more than once, so it's just $$6^3 = 216$$.

Honestly, if I had to pick one of these to discount, it would probably be the first one -- and yet that's the "correct" "answer" to this textbook problem. Am I wrong in being disgusted by what a poor, ambiguous, useless textbook problem this is?! How could anyone in their right mind assign 18 as the correct answer and say the other three are incorrect?! Am I missing something that makes the first interpretation the only one possible?

[[1 ]] Lack of Clarity :

Well , YES , the Question could be written with a little more clarity , since it can be taken in various ways.

[[ 2 ]] "Am I missing something that makes the first interpretation the only one possible?"

Well , YES : Check the high-light to see why $$18$$ was the "Correct" Answer given by the Author :

"Suppose registrants at a large convention are offered 6 sightseeing tours on each of 3 days. In how many ways can an attendee arrange to go on a [[ SINGLE ]] sightseeing tour [[ SINGULAR ]] planned by this convention?"

Now the "Correct" Answer : We have $$6 \times 3 = 18$$ tours & we can choose $$1$$ in $$18$$ ways.

[[ 3 ]] Slightly More Clarity :

"Suppose registrants at a large convention are offered 1 free sightseeing tour. There are 6 sightseeing tours on each of 3 days. In how many ways can an attendee arrange to go on a free sightseeing tour planned by this convention?"
[[ the BOLD is not necessary in the textbook , I am using it here to high-light what the CRUX is ]]

We can then easily make out that $$18$$ is the "Correct" Answer.

• I suppose I can agree there is some tendency for singular phrasing here, but then I would add that in this case, there is a further disqualifying feature: the question does not test any conceivable principle of statistics or probability. It's just a poorly phrased statement that there are 18 sightseeing tour slots, and then you are to say, "There are 18 sightseeing tour slots." Commented Oct 23, 2023 at 4:36
• You are right about that Poor Wording , though this was in "Early Chapter" , hence it might be setting the stage for later higher topics !
– Prem
Commented Oct 23, 2023 at 7:03