Amongst 100 people who were surveyed 90like Amitabha,77 like Abhishek,75 Likes Hrithik and 70 like Shahrukh. a)Minimum number of people who like all 4. b)Maximum number of people who like all 4. Pls explain using minimum overlapping line method

For Max , we can use the trick E1+e2+e3+e4=100. where e1 =exactly one who exactly like only Amitabh or Abhishek or Hrithik or Shahrukh. Similarly e2= exactly two who like two people E3 who like 3 e4 who like all four… Also e1+2e2+3e3+4e4=90+77+75+70=312 …. Double counting If we subtract above equation we get e2+2e3+3e4=212 We want e4 who like four to be max hence we want e2+2e3=minimum. We want rhs to be multiple of 3. So e2+2e3=2 Hence 3e4= 212-2=210 Hence e4(max)=210/3=70 For minimum who like all 4——— Amitabh =90. Hence No of people who didn’t like Amitabh out of 100=100-90=10 Similarly for Abhishek 100-77=23 Similarly for Hrithik 100-75=25 Similarly for Shahrukh 100-70=30 Hence Total =10+23+25+30=88 Hence minimum who like all 4 = 100-88=12