# Venn diagram and elementary set theory

Here is a question,

There are 70 students in a music class. 20 can play violin. 10 can play the flute. 15 can play the guitar. 4 can play both violin and flute. 7 can play both flute and guitar. 3 can play all violin flute and guitar. 9 have just enrolled into the class and can not play any instrument fluently.

1. How many people can play both violin and guitar?
2. How many people can play only flute and no other instrument?

(It should be solved by making Venn Diagram.) When I tried, I was able to do the first, the answer comes out to be 0. But I am stuck with the second one, I tried 10-7-3-4 but the answer comes out to be negative. Please help me out.

EDIT: Here is the diagram,

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Could you display the Venn diagram you've drawn? Note for instance that among the 7 that play flute and guitar there are 3 that also play violin, so you shouldn't subtract them twice by writing -7-3 –  t.b. Nov 3 '11 at 6:37
$10-7-4+3$ is what I should be doing? –  Fahad Uddin Nov 3 '11 at 6:46
Wait a minute: your numbers do not add up: I count 68 = 9+20+15+4+3+7+10 students in your class, not 70. Maybe there's a translation error and I misunderstood your solution. Are the descriptions of what instrument(s) the students play meant to be exclusive? Please check this and the numbers again. –  t.b. Nov 3 '11 at 6:51
I find the numbers hard to interpret, since they don't add up to nearly enough. I imagine we are supposed to assume that the total who can play at least one the instruments is $61$. In a standard problem of this type, the given numbers $20$, $10$, $15$, $4$, $7$, $3$, $9$ should add up to more than $70$, because of the overlaps. But they only add up to $68$. –  André Nicolas Nov 3 '11 at 6:52
Its a class problem developed by our teacher. This may be a error by her side. –  Fahad Uddin Nov 3 '11 at 7:10

Start with the intersection of the three circles in the Venn diagram. There are 3 people that can play all instruments. This leaves 1 (=4-1) person who can play violin and flute, but not the guitar. Similarly of the 7 that can play flute and guitar, 3 can also play flute so that leaves 4 that can play flute and guitar only. Since 10 people play that flute that leaves 2 people that can only play the flute.

That leaves you with three spots left in your Venn diagram. You have equations for

1) The total number of people (don't forget the 9 that don't play anything!) 2) The total number of guitar players 3) The total number of violin players

Solve this simultaneously (although I am getting some nonsense answers out so maybe I am doing something silly!)

Edit: As the comments point out, there does appear to be something funny about the problem.

With regards to the diagram you posted Akito, I interpret the problem as (for example) 20 people in total that can play the violin. Thus you shouldn't be writing 20 where you have - you should have (what you have written as) 20+4+3+0 as the total number of violin players (i.e. 20)

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Thanks. I will change it to $13$ –  Fahad Uddin Nov 3 '11 at 7:09