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
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.
Then:
22 * 2 = 44assigning everyone two composers to start with (as per Statement A).
And:
17 + 15 + 15 = 47the total number of composers chosen.
Therefore:
47 - 44 = 3composers are left over and must be assigned to separate students (as per Statement B) and the answer is 'c) 3 students'
Additional:
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:
d)
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.
a)
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.
c)
As we just saw 3 students is a possible answer, but lets check the others.
e)
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.
b)
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.
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
