A problem on minimization

ans is 2 but how?????

@ srk, ans is 2.

@ Anant sir, - I think it can also be got from what I did. If the last part of my soln is unclear, then I'm elaborating it here.

As already said, in the union of A,B,C there must be 2 elements that are not present in any of the sets, let me set set A as my reference.

Let these "new" elemenets be x and y.

Also let m of these elements be in B and n in C.

CASE -1, m>n.

(m,n)=(2,1).

So that gives \left|A\cup C \right|=100.

\left|A\cup B \right|=98 & \left|C\cup B \right|=99.

Also \boxed{ \sum{\left|C\cup B \right|}=297}....(1).

CASE-II m=n.

a) x in B, y in C.
b) x,y present in both sets B and C, that in the previous fashion gives a) \sum{|A\cap B|}>297 & for b) \sum{|A\cap B|}=297.

CASE 3 when n>m. (n,m)=(2,0) (2,1) - in similar fashion we have for (n,m)=(2,0) \sum{|A\cap B|}> 297 & for (n,m)=(2,1) \sum{|A\cap B|}= 297.

Thus minimum {|A\cap B \cap C|}= 297-\left(\sum{|A|} \right)+\sum{|A\cup B\cup C|} =98.

Now pls reply, whether correct or not.

IF x,y belongs toR AND x²+y² +xy=1,then minimum value of

x³y+xy³+4=?

Sorry that's wrong sign in my previous post. Editing.

I have the following:
The given condition n(A)+n(B)+n(C)=n(AUBUC) translates to the equation
2¹⁰¹ + 2^|C| = 2^|AUBUC|
That is
1 + 2^|C|-101 = 2^|AUBUC|-101
Since LHS > 1, an odd number and same as a power of 2, we must have |C| =101 and hence |AUBUC| = 102.

Next, we have from PIE, that
|A\cup B\cup C| = \sum_\mathrm{cyc}|A|-\sum_\mathrm{cyc}|A\cap B|+|A\cap B\cap C|
Further,
|A\cap B|=|A|+|B|-|A\cup B|
and similarly for the other two intersections, so we get
|A\cap B\cap C| = |A\cup B\cup C|+(|A|+|B|+|C|)-\sum_\mathrm{cyc}|A\cup B|
Further, since A\cup B \subset A\cup B\cup C, so |A\cup B|\le |A\cup B\cup C|
and similarly for other two unions.
Hence, we get
|A\cap B\cap C|\ge |A|+|B|+|C|-2|A\cup B\cup C|=97
So, |A\cap B\cap C| is bounded below by 97.

The question is whether or not the equality be attained.

it should be 98

kaymant Sir please tell the correct answer

Soumik check my given sets and find error ?
i am again getting 98

consider
set A={1,2,3,4........100}
set B={3,4,5,6........102}
set C={1,2,3,4........101}