100 freshmen were interviewed. 28 took PE,31 took BIO, 42 took ENGLISH, 9 took PE and BIO, 10 took PE and ENGLISH, 6 took BIO and ENGLISH, 4 took all three subjects. how many students took none of the three subjects?, how many students took PE but not BIO and ENGLISH?, how many students took BIO and PE but not ENGLISH?



I hope you are familiar with Venn diagram....this is a very useful tool in solving this type of problems..i just cant draw  3 intersecting circles here to illustrate the diagram..but as u find how to use it..start filling up the center of the intersecting 3 circles with the element common to them (that is 4 in this case)...then expand outwardly from the center..and proceed to the elements where only 2 categories share( 6, 10 and 9 in this case).

And you can get all the information needed easily..
Students who took none of the 3 subjects = 20
Students who took PE but not BIO and ENGLISH = 16
Students who took BIO and PE but not English = 2 
3 3 3