Question 1151779
i think it's going to work like this.
you want to break each category up into different subgroups.
start with the students that are leaning all 3.
that's an only category, since it includes only students that are taking all 3.
then work up to the category of pairs of courses.
to make them pairs of courses only, you have to subtract the students who are taking all 3.


there are 10 students who are taking arabic and yoruba.  
included is the students that are taking all 3.
subtract 3 from this category and you have 7 students who are taking arabic and yoruba only.


there are 8 students who are taking arabic and french.
included is the students that are taking all 3.
subtract 3 from this category and you have 5 students who are taking arabic and french only.


there are 5 students who are taking french and yoruba.
included is the students that are taking all 3.
subtract 3 from this category and  you have 2 students who are taking french and yoruba only.


what you have now is:
28 taking arabic
30 taking french
42 taking yoruba
3 taking all 3 only.
7 taking arabic and yoruba only.
5 taking arabic and french only.
2 taking french and yoruba only.


from the 28 who are taking arabic, you have to subtract 3 taking all 3 only and 7 taking arabic and yoruba only and 5 taking arabic and french only.
this leave you with 28 - 3 - 7 - 5 = 13 who are taking arabic only.


from the 30 who are taking french, you have to subtract 3 taking all 3 only and 5 taking arabic and french only and 2 taking french and yoruba only.
this leaves you with 30 - 3 - 5 - 2 = 20 who are taking french only.


from the 42 who are taking yoruba, you have to subtract 3 taking all 3 only and 
7 taking arabic and yoruba only and 2 taking french and yoruba only.
this leave you with 42 - 3 - 7 - 2 = 30 taking yoruba only.


what you have now are:


only arabic = 13
only french = 20
only yoruba = 30
only arabic and french = 5
only arabic and yoruba = 7
only french and yoruba = 2
only arabic and french and yoruba = 3


this adds up to 80 students which means 20 students aren't studying any language at all since there are 100 students total.


you can work your way back up to the original figures by adding back the sub categories into their respective categories.


for example:
13 students took arabic only.
add 5 to include the students who took arabic and french only.
add 7 to include the students who took arabic and yoruba only.
add 3 to include the students who took all 3.
you get a total of 13 + 5 + 7 + 3 = 28 students who took arabic.
5 took arabic and french only.
add 3 to include the students who took all 3.
you get a total of 5 + 3 = 8 students who took arabic and french.


a generalized formula from the original numbers would be.


number who took arabic minus number who took pairs of courses that include arabic plus number who took all 3 = number who took arabic only.
this would be 28 - 8 - 10 + 3 = 13


number who took french minus number who took pairs of courses that include french plus number who took all 3 = number who took french only.
thi would be 30 - 8 - 5 + 3 = 20


number who took yoruba minus number who took pairs of courses that include yoruba plus number who took all 3 = number who took yoruba only.
this would be 42 - 10 - 5 + 3 = 30.


working your way down, you would then subtract number who took all 3 from number who took pairs of courses.

for example, number who took arabic and french minus number who took all 3 = 7.


same idea for he others.


i made a chart which summarizes what was said above.
what it looks like is shown below.


<img src = "http://theo.x10hosting.com/2020/012901.jpg" alt = "$$$" >


in any case, you solution should be:
(1). How many student where studying no language
20 students were studying no language.


(2). How many student had yoruba as their only language
30 students has yoruba as their only language.


(3). How many student study french if and only if study yoruba
not sure how to interpret this one.
i think this would be the students who study french and yoruba only.
i believe this number is equal to 2.
i don't think this would include the students who studied all 3 since that would include the students who studied arabic as well.