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The number of pairs of set (X, Y) are there that satisfy the condition X, Y ⊆ {1, 2, 3, 4, 5, 6} and XY = Φ ________.

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Here  X and Y are representing subsets such that X ∩  Y  =  Φ meaning that X and Y are disjoint subsets..So for finding number of such pairs which are disjoint to each other and let the size of the set = n.

Let we have number of elements in X  =  r..

So number of ways of selecting elements in X such that it contains 'r' elements  =  nCr

So we have number of elements left for inclusion in Y =  n-r..

Now each of these 'n-r' elements can be either selected or rejected , hence 2 choices for each element..

Hence total number of possibilities of Y such that it is disjoint to X  =  2 . 2 . ......(n-r) times

                                                                                                  =  2n-r

Hence number of pairs (X,Y) such that X contains 'r' elements         =  nCr  *   2n-r

Now 'r' varies from 0 to n..

Hence total number of pairs (X,Y)          =         Σ  nCr  *   2n-r

                                                           =        (1 + 2)n

                                                           =         3n

But (X,Y) is same as (Y,X)  , hence double counting is to be avoided..

Hence number of such pairs                  =       [ ( 3n  -  1 ) / 2 ] + 1

Substituting n = 6 , we get number of pairs     =   728 / 2  + 1

                                                                   =   365

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