Solve and explain the answer

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A set X=1,2,3,4,5To ifnd: The number of different ordered pairs (Y,Z) such that YX,ZX and YZ=ϕ.Since, YX,ZX, hence we can only use the elements of X to construct sets Y and Z.
n(Y) Number of ways to make Y Number of ways to make Z such that YZ=ϕ
0 C05 32
1 C15 16
2 C25 8
3 C35 4
4 C45 2
5 C55 0

Let us explain anyone of the above 6 rows ray third row. In third row,
Number of elements in Y=2
Number of ways to select Y=C25 ways.
Because any 2 elements of X can be part of Y.
Now, if Y contains any 2 elements, then these 2 elements cannot be used in any way to construct Z because we want Y U Z = ϕ. And from the remaining 3 elements which are not present in Y, 23 subsets can be made each of which can be equal lo Z and still Y ⋂ Z = ϕ will be true.  Hence, total number of ways to construct sets Y and Z such that Y ⋂ Z = ϕ

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