# How to find the number of elements between three different sets when there is not a total sum of their elements?

The problem is as follows:

In an institution they offer three language courses one being German, the other French and the last one Polish. Four students enrolled in the three courses, six students in Polish and German and seven in French and Polish. If all students enrolled in Polish also enrolled in German or French. How many of the students were in the Polish course?

The existing alternatives in my book are:

• 9

• 7

• 6

• 5

• 8

What I tried to do is to build up a Venn diagram as shown below:

But from this point I am stuck as I do not know how to relate the number of students in Polish language since there is not known the total number of elements from all sets together.

Can this problem be solved without needing this information?.

There is one thing regarding how I understood the problem as it mentions seven students enrolled in French or Polish so by interpreting this information I assumed that $P=7$ and $F=7$ therefore the diagram would become into this:

Edit:

By reviewing what it was mentioned in the problem I noticed that earlier assumptions did considered only elements belonging to $P$, $G$ or $F$ but it was not the case, therefore I changed this approach and "calculated" the elements for $G$ and $P$ and $G$ and $F$ alone being them $2$ and $3$ respectively.

This can be seen in the figure below:

But this is how far I went in the problem. How can I take it from here to reach the solution?.

Therefore I'm stuck at this, can somebody help me to go in the right track or what conclusion I have to take to solve this problem?

• Your final attempt is correct. There is an implicit assumption here that a student who is enrolled in German and Polish may also be enrolled in French and that a student who is enrolled in French and Polish may also be enrolled in German. – N. F. Taussig Apr 10 '18 at 23:38
• @N.F.Taussig I totally missed the clue regarding the number of students in Polish alone being none as it is mentioned in the problem they also took other courses from this on the problem can be solved. :) – Chris Steinbeck Bell Apr 11 '18 at 18:22

You know that there is no student who solely studies polish. So the answer is $6+4+7$.

Edit: as you know that there are $6$ students in G and F altogether. Now $4$ of them are in all three courses, so there are $2$ left which solely study German and Polish. Similar, only three student are studying French and Polish solely. Hence the numbers are $2$, $4$ and $3$ instead of $6$, $4$ and $7$ so the final answer is $2+4+3=9$.

• I've added the existing alternatives which are available on my book and none seems to match with what you say as the answer. So is my book wrong?. – Chris Steinbeck Bell Apr 9 '18 at 15:14
• This solution seems to assume that the problem said "... and seven in French and Polish" -- but what it actually said, according to the OP's quote, was "... and seven in French or Polish". – hmakholm left over Monica Apr 9 '18 at 15:38
• @Michael Hoppe Maybe can you explain a little bit better?. Which part of my drawing is not correct?. I thought that the intersection between $G$ and $P$ is $6$. How do I know that there are $6$ between $G$ and $F$. To come to that conclution I may had to establish what it is solely on $P$ which it is what it is being asked in the first place. Maybe can you include a Venn diagram? – Chris Steinbeck Bell Apr 9 '18 at 17:02

Your first Venn diagram is wrong, the second one is correct. We are told that there are "Four students enrolled in the three courses" so the middle portion where all 3 sets overlap has the number 4. But the "six students in Polish and German" includes those who are enrolled in all 3 courses. The overlap between Polish and German should have the number 6- 4= 2, not the full 6. Similarly, the "seven in French and Polish" Includes the 4 who are taking all three. The overlap of French and Polish only should have the number 7- 4= 3. We are told that "all students enrolled in Polish also enrolled in German or French" so the number in the Polish only set is 0. There are a total of 4+ 2+ 3= 9 students taking Polish.

• Is it just me or my computer is not displaying your paragraph correctly. It looks as if you tried to make bold one word but it is not showing it that way. As you mentioned yes it took me a while to notice that error because they did not used the word "only" hence I had to understand that the numbers given included the $4$ so had to subtract from them. The last clue was not very obvious to me but as they said all Polish enrolled also in other courses there was no people who took Polish alone. The rest is as you mentioned. Thanks! – Chris Steinbeck Bell Apr 11 '18 at 18:21
• Yes, I should have used ** rather than [b]. Unfortunately that varies from one board to another. – user247327 Apr 11 '18 at 20:50

"Four students enrolled in the three courses,"

This is unambiguous. The students who enrolled in all three classes is $4$.

Venn diagram is so far: "six students in Polish and German"

This IS ambiguous.

It can be interpretted in 3 ways:

(Most likely): $6$ students are taking both polish and german. Some of them might be taking French as well. The Venn Diagram for that is: (Less likely but possible): There are $6$ students who are taking both polish and german but are not taking French. The Venn Diagram for that is: (Very unlikely): If you consider all the students who are taking Polish or are taking German there are $6$ such students. (This is very bad language usage). The Venn diagram is: and seven in French and Polish.

Ditto.

More Diagrams:   If all students enrolled in Polish also enrolled in German or French.

THe means there were no students in polish alone.

In the most likely event we have this final diagram: In this case we can see the students taking polish is $0+3+4+2 = 9$. (I am absolutely certain this was the intended answer).

In the less likely event we have this final diagram: There we can see the students taking polish are $0+7+4+6 = 17$. (I am fairly certain this was not the intended answer).

And in the least likely even we have: From that we really can't determine anything. We have $4$ students taking all three so there are there are $2$ other students taking either German or Polish and $3$ other students taking either French or Polish. If the $2$ students are both taking German and not polish then there are $3$ taking french only and there are only $4$ taking polish. If one of the other $2$ students is taken polish and the other taken german but not polish, then there are $2$ students taking french but not polish and there are $5$ students taken polish. If both the $2$ other students are both taking polish, then there is $1$ taking french but not polish and there are $6$ taking polish.

(I am absolutely certain this is not the intended answer)

• I also noticed during a closer examination of this problem that there were more interpretations. But as you indicated the intended answer was $9$. The two diagrams were a nice addition. – Chris Steinbeck Bell Apr 13 '18 at 2:02