In a school where there was a compulsion to learn at least one foreign language from the choice given to them, namely German, French and Spanish. Twenty eight students took French, thirty took German and thirty two took Spanish. Six students learnt French and German, eight students learnt German and Spanish, ten students learnt French and Spanish. Fifty four students learnt only one foreign language while twenty students learnt only German. Find the number of students in the school.

Exactly 1 subject = a+b+c ---> Represented by X

Exactly 2 subjects=d+e+f -----> Represented by Y

Exactly 3 subjects= g -----> Represented by Z

So X + Y+ Z + none = total --------------> (I)

German + French + Spanish = (a+b+c) +2(d+e+f) + 3(g) = X+ 2Y + 3Z -----------> (II)

So X+2Y+3Z= 30+28+32=90 

Given X = 54

So 2Y +3Z = 36--------> (1)

Given ,

French and German = 6 => d+g = 6

German and Spanish = e+g = 8

French and Spanish = f + g = 10

adding all the three (d+e+f) + 3g = 24

Y + 3Z = 24 ------> (2)

solving 1 and 2 you get Y=12 and Z=4

Therefore Total = X+Y+Z+None = 54+12+4=70