In how many ways can two distinct subsets of the set `A` of `k(kgeq2)` elements be selected so that they haves exactly two common elements?

Explanation

Let two subsets of set A be B C.

There are exactly two elements in `B cap C`.
Two elements can be selected in `""^(k)C_(2)` ways.
Now each of the remaining (k-2) elements can be put in any of the three regions X, Y or Z. [Here, Z represents `(A cup B)']`
Number of ways in which this can be done are `3^(k-2)`.
But we must exclude the case in which all the elements are put in region Z as otherwise sets B and C will be identical.
Also, the order of subsets should not be considered.
Hence, total number of ways `=(""^(k)C_(2)(3^(k-2)-1))/(2)`