in a survey of 25 students it was found that 15 had taken mathematics,12 had taken physics and 11 had taken chemistry,5 had taken mathsematics and chemistry,9had taken mathematics and physics,4 had taken physics and chemistry and 3 had taken all three subjects.find the number of students who had taken-

- at least one of the three subjects
- only one of the three subjects

M = a +e +c +b = 15

C = f +e + c + d = 11

P = b + c + d + g =12

Maths and chemistry = e + c =5

Maths and physics = b + c = 9

Physics and chemistry = d +c = 4

All three = c = 3 (1)

So using (1) in above three equations, we get

e = 2 , b = 6 ,d = 1

Hence using these three values in the above equation,we get

a +e +c +b = 15

Or a = 15 -2-3-6 = 4

f +e + c + d = 11

Or f = 11 - 2-3 -1 = 5

b + c + d + g =12

So g = 12 -6 -3 -1 = 2

So Number of student who have taken atleast one subject = a + b + c + d +e+ f + g = 4 +6 +3 +1+2+5 +2 = 23

And number of student who have taken one subject only = a+f+g = 4 + 5 + 2 = 11

**
**