in Mathematical Logic retagged by
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18 votes
18 votes

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?

  1. $(a, b, c) = (0.51, 0.51, 0.51);$
  2. $(a, b, c) =(0.61, 0.71, 0.67);$
  3. $(a, b, c) = (0.68, 0.68, 0.68);$
  4. $(a, b, c) = (0.49, 0.49, 0.49);$
  5. None of the above.
in Mathematical Logic retagged by
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3 Answers

32 votes
32 votes
Best answer
$6$ preference order for voter are possible:

$ABC,ACB,BCA,BAC,CAB,CBA$

Also given that

$a=ABC+ACB+CAB(A$ prefer over $B)   ---(1)$

$b=BCA+BAC+ABC(B$ prefer over $C)  ---(2)$

$c=CAB+CBA+BCA(C$ prefer over $A)  ---(3)$

Adding $1,2$ and $3$ we get

$a+b+c=2(ABC+BCA+CAB)+ACB+BAC+CBA$

Now we know that $ABC+ACB+BAC+BCA+CAB+CBA=1$ therefore

$[ABC+ACB+BAC+BCA+CAB+CBA]<[2(ABC+BCA+CAB)+ACB+BAC+CBA]<2(ABC+ACB+BAC+BCA+CAB+CBA)$

Hence we can say that value of $a+b+c$ must be between $1$ and $2$

Option $(C)$ value greater than $2$.

Hence correct answer is $(C)$.
edited by

4 Comments

@Shaik Masthan

Can a + b + c = 2 ?

or is (a + b + c) strictly less than 2?

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on what basis is the equation written . I could not understand . can any one explain it
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@Ruchi Vora

fraction a of the voters prefer Amar to Birendra

It means all the cases where Amar should come before Birendra  i.e. ABC or CAB or ACB (position of C doesn't matter)

so thats why a = ABC+CAB+ACB

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3 votes
3 votes
I will just answer for option (C)  , rest 3 would be then easy to derive.

LET THERE BE ONLY 100 PEOPLE .
(TRY TO DRAW A Venn diagram FOR MY EXPLANATION, IT WOULD BE SIMPLE TO UNDERSTAND)

Amar=  Modi , Birendra = Kejriwal , C= Rahul

GIVEN THAT :
Any 68 people say (Modi > Kejriwal) (means Modi is greater than Kejriwal)
Any 68 people say (Kejriwal > Rahul)

But we have only 100 people. So there is definitely some intersection.
So at least (68+68-100)=36 people made both the statements (i.e Modi > Kejriwal and Kejriwal > Rahul means these 36 people actually mean Modi > Kejriwal).

So these 36 people can NEVER Contradict themselves and make the 3rd statement : "Rahul > Modi" .

So only at most remaining 100-36 = 64 people can make the 3rd statement but option (C) says 68 people made this statement which is NEVER possible as discussed above.

2 Comments

yaha bhi bhakti !! 😂
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Mujhe to isse hi samaj aaya! 😎
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0 votes
0 votes

Let $x$ be the total number of voters

Let $U$ be the Universal set of voters, $A$ be the set of voters who vote for Amar, $B$ be the set of voters who vote for Birendra, $C$ be the set of voters who vote for Chanchal

voters prefer $P$ to $Q$ means voters vote for $P$ but not for $Q$

$\therefore$ Answer is Option $\LARGE C$

Answer:

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