What is impossible?

Question

Hi recently i appeared in an aptitude,there was a problem that i realy cant understand please provide some idea, how to solve it.(and sorry to for poor English.)

(Question)-> Three candidates, Amar, Birendra and Chanchal stand for the local election. Opinion polls are conducted and show that fraction a of the voters prefer Amar to Birendra, fraction b prefer Birendra to Chanchal and fraction c prefer Chanchal to Amar. Which of the following is impossible?

(a) (a, b, c) = (0.51, 0.51, 0.51);
(b) (a, b, c) = (0.61, 0.71, 0.67);
(c) (a, b, c) = (0.68, 0.68, 0.68);
(d) (a, b, c) = (0.49, 0.49, 0.49);
(e) None of the above.

Answer 1

If you tried to list of possible preferences people can have are either

1. ABC (means you prefer A to B, prefer B to C and therefore also prefer A to C)
2. ACB
3. BAC
4. BCA
5. CAB
6. CBA

in this case you'll find that each fraction of the population represents:

1. a=ABC+ACB+CAB
2. b=ABC+BAC+BCA
3. c=BCA+CAB+CBA

therefore a+b+c = 2(ABC+BCA+CAB)+ACB+BAC+CBA as you notice not all groups within the population are repeated. we can therefore assume than (a+b+c) can never be more than twice the population since each member of the population is represented twice at the most.

out of the options C is the one where the sum is more than 2. and is therefore the impossible value.