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)
There are a couple of ways to approach the problem; I’ll start with a very straightforward one from first principles. First, here’s a picture:
The blue part of the diagram represents those students who do not participate in soccer, basketball, or track. We’re interested in $x$, the percentage who play all three sports. One way to solve the problem is to fill in the Venn diagram in terms of $x$. I’ve started: we know, for instance, that $2\%$ play soccer and basketball, and $x\%$ play all three sports, so $2-x$ must be the percentage playing soccer and basketball but not track. The figures $6-x$ and $9-x$ were obtained by similar reasoning. Now go after the percentage playing just soccer, just basketball, and just track. We know that $21\%$ play soccer, and $(2-x)+x+(6-x)$ percent play soccer and at least one other of the three sports, so what percentage (in terms of $x$) play just soccer? Repeat this procedure with basketball and track to fill in the last three regions of the Venn diagram. Finally, add up the percentages of all of the white regions; that total, which will involve $x$, must be the $100-36$ percent who play at least one of the three sports, and you’ll have a simple equation to solve for $x$.
If you know something about calculating the cardinalities of unions and intersections of sets, you can short-circuit this calculation. Let $S,B$, and $T$ be the sets of students playing soccer, basketball, and track, respectively. Pretend that there are $100$ students altogether, so that percentages are simply numbers of students. Then you want $|S\cap B\cap T|$, and you know $|S|$, $|B|$, $|T|$, $|S\cap B|$, $|S\cap T|$, $|B\cap T|$, and the cardinality of the complement of $S\cup B\cup T$. That last figure gives you $|S\cup B\cup T|$. Now, do you know a formula that relates all of these quantities? If so, you’re in business, because you know all but one of them.
Added: However, the problem is faulty, because if you actually carry out the computations, which I unfortunately did not bother to do when I originally posted the answer, you’ll find that $x$ is negative!
Best Answer
The given problem is ill-posed.
Since 35% play football and 25% play cricket, then those who play neither can only comprise between 40% ($F$ and $C$ are disjoint) to 65% ($C \subseteq F$) of the students. This contradicts the other given piece of information (20% play neither).