The 22 students of the course of History of Italian Opera must write a short paper about at least two composers, they have a better grade, if they choose to write a paper about one more composer. among the students 17 choose Verdi, 15 choose Rossini, 15 Donizetti. how many students have chosen to write three papers?

a) 2 students
b) it is impossible to answer this question
c) 3 students
d) nobody has chosen three composers
e) 15 students

  • 3
    $\begingroup$ Welcome to Puzzling! This is a site for discussion and sharing of puzzles. Puzzles can be mathematical in nature, but this appears to be homework or even a test. $\endgroup$
    – Deusovi
    Jul 11, 2016 at 10:57

3 Answers 3


Assume Statement A:

All 22 fulfilled the requirements and chose at least two composers.

Statement B:

The lack of resultant choices also means no student answered with more than three.


22 * 2 = 44 assigning everyone two composers to start with (as per Statement A).


17 + 15 + 15 = 47 the total number of composers chosen.


47 - 44 = 3 composers are left over and must be assigned to separate students (as per Statement B) and the answer is 'c) 3 students'


To prove there is at least one possible combination: 3xVDR, 7xVD, 7xVR, 5xDR. Where the capital letters represent the composer by relevant initial.


My answer:

first we can see that 17 +15 +15 = 47, and 22*2 = 44. 47 - 44 = 3. As 3>0 there is at least one student that took more than 2 composers.


It's obvious that Hermione Granger would take 4 composer. However there only are 3 different composers chosen, so it's impossible to take more than 3. At least 3 students have taken 3 composer.


As we just saw 3 students is a possible answer, but lets check the others.


We are talking about students so some of them won't have the work done. if 5 students haven't done their homework, we have 17 "good students". Among them we could have 15 that chose 3 composer and 2 only 1 for a total of: 15*3+2*1=47 composer. OK that's possible.


As we explained both c and e are possible answers. From here we can take 2 different paths: say only b is correct as we can't find the correct answer between c & e, or say that both c & e are correct (that's to say b is incorrect).

I would choose:

answer b.

  • $\begingroup$ There might be 3 students who have chosen three composers. $\endgroup$
    – user26302
    Jul 11, 2016 at 11:56
  • $\begingroup$ yes, that's what I am saying, but that doesn't make it the answer $\endgroup$
    – Sechiro
    Jul 11, 2016 at 12:03
  • 2
    $\begingroup$ But... if you choose b then you have answered the question and b is wrong ! $\endgroup$
    – Fabich
    Jul 11, 2016 at 12:48
  • $\begingroup$ damn, that's true +1. But I don't want to modify it, I like nonsense. $\endgroup$
    – Sechiro
    Jul 11, 2016 at 12:56

c) 3 Students

The minimum papers written (2 each student) is 2·22=44

But we have 17+15+15= 47 papers wrottem, so we have 47-44=3 students who wrote 3 papers

We can also know the exact distribution of papers:

If 3 students wrote 3 papers, then 19 S (students) wrote only 2 papers. If we take 3 papers from each compositor, we keep 14 V (Verdi) 12 R (Rossini) and 12 D (Donizetti). If 14 S wrote about Verdi, then 5 didn't, so 5 S wrote about Rossini and Donizeti, leaving: 14 S, 14 V, 7 R and 7 D. So, 7 S wrote about Verdi and Rossini and 7 about Verdi and Donizetti.

In summary:

3 Students wrote about Verdi, Rossini and Donizetti
5 Students wrote about Rossini and Donizetti
7 Students wrote about Verdi and Rossini
7 Students wrote about Verdi and Donizetti


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