Question

An institue conducts scholarship exams for Maths,Science and English.In total 280 students study Maths,254 students study Science and 280 students study English.97 students study both Maths and Science,138 students study both Science and English,152 students study both Maths and English.73 students study all the three subjects.How many students are there at the institute?

Answer 1

Please refer to the attached Venn Diagram.

Students who study only Maths     = 280 - 79 - 73 - 24 = 104
Students who study only Science  = 254 - 24 - 73 - 65 = 92
Students who study only English   = 280- 79 - 73 - 65  = 63

Total number of students = 104 + 92 + 63 + 79 + 73 + 24 + 65 = 500

So the total number of students in the institute is 500.

Answer 2

Let us represent the sets of students by the first letters M, S, E of the subjects.  The cardinalities of the sets are as in:

N(M) = 280                 N(S) = 254             N(E) = 280
N(M+S)= 97               N(S+E) = 138         N(M+E) = 152        N(M+S+E) =  73

The total number of students in the University 
   = N (Union of sets M, S & E)
   = N(M) + N(S) + N(E) - N(M+S) - N(S+E) - N(M+E) + N(M+S+E)     Formula
   = 280 + 254 + 280 - 97 - 138 - 152 + 73
   = 500