A class got 48 distinctions in maths, 25 in physics and 30 in chemistry. If these went to a total of 68 students and only 5 students got distinctions in all the three subjects, how many students got distinctions in exactly two of the three subjects?

Asked by Topperlearning User | 4th Jun, 2014, 01:23: PM

Expert Answer:

Let M, P and C denote the set of students who got distinction in maths, physics and chemistry respectively.
Then n(M) = 48, n(P) = 25, n(C ) = 30
n(M  P C) = 68 and n(M  P  C) = 5
      Therefore,
n(MP C) =  n(M) + n(P)  + n(C) – n(MP) – n(MC) - n(PC) + n(MPC)
     68         = 48 + 25 + 30 – n(M  P) – n(M  C) - n(P  C) + 5 
 n(M P) + n(M C) +  n(P C) = 108 – 68 = 40
Now consider the following Venn diagram:
 
 
Here,
a denotes the number of students who got distinction in maths and physics only.
b denotes the number of students who got distinction in maths and chemistry only.
c denotes the number of students who got distinction in physics and chemistry only.
d denotes the number of students who got distinctions in all the three subjects.
Thus,      d = n(M  P  C) = 5 and
a + d + b +d + c + d = 40
Therefore, a + b + c = 25

Answered by  | 4th Jun, 2014, 03:23: PM