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Ten eggs are distributed among $\text{ABCD}$ in ratio $1:2:3:4$ randomly. It is known that $\text{A}$ gets less eggs than $\text{B,}$ and $\text{C}$ gets more eggs than $\text{D}.$ If $\text{A}$ gets half the number of eggs of $\text{B},$ then which of the following is necessarily true ?

  1. $\text{C}$ gets an even number of eggs
  2. $\text{D}$ gets an even number of eggs
  3. $\text{C}$ gets an odd number of eggs
  4. $\text{D}$ gets an odd number of eggs
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10 eggs distributed among A,B,C,D in the ratio 1:2:3:4 not necessarily in that order.

Since, $1+2+3+4 = 10,$ the $10$ eggs must be distributed as $1,2,3,4.$

Let us assume ,

No of eggs  A gets =p

no of eggs B gets =q

no of eggs C gets=r

no of eggs D gets=s .

we can think it like a bijection function from {p,q,r,s} ->{1,2,3,4}

now it is given ,

  1. p<q
  2. s<r

now it is  also given ,

p=q/2

so from this we can infer that q is either 4 or 2 as they both divide by 2 because it is guaranteed that p got some distributions of eggs.

case 1:- q=4

        so, p=2                     [ because p=q/2]

              r=3

              s=1                      [ because s<r]

case 2:- 

                q=2

                p=1                    [because p=q/2]

                r=4                      [because s<r]

                s=3

so this two are the possibility and we from can see here ,

C can have both even or odd number of eggs both are possible .

But D has to take necessarily odd number of eggs .

The two bijection can look like :- 

 

            

So correct option is D.        

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