GATE Overflow Test Series | Mock GATE | Test 6 | Question: 62

Question

Consider set $A = \{1,2,3,4\}.$ From all the subsets of $A,$ we define a new set $X$ as
$X = \{(P,Q) \mid |P \Delta Q| = 2\},$ where $(P,Q)$ is ordered pair $(P,Q)$ and $\Delta$ denotes the symmetric difference operation. The number of elements in $X,$ i.e., $|X|$ is ________

Comments

as it's given that (P,Q) are ordered pairs we why aren't we taking 96@8 are final answer as (Q,P) is also an ordered pair

Answer 1

Both $P$ and $Q$ are subsets of $A.$ There are three choices which we can consider in counting.

1. Both the elements of symmetric difference belong to set $P.$ Then there can be $\binom{4}{2} \times 2^2$ possibilities as the exclusive values for set $P$ can be chosen in $\binom{4}{2}$ ways and $Q$ can be any of the subsets of remaining three elements (elements in these subsets are also present in $P)$ which can be in $2^2$ ways.
2. Both the element of symmetric difference belong to set $Q.$ Similar to above we get $\binom{4}{2} \times 2^2$ possibilities.
3. One element of symmetric difference come from $P$ and other from $Q.$ Then there can be $\binom{4}{1} \times \binom{3}{1} \times 2^2$ possibilities ($2^2$ for the subsets of the remaining $2$ elements which are present in both $P$ and $Q).$

$\therefore$ The number of elements in $X,$ i.e., $|X| = 2 \times \binom{4}{2} \times 2^2 + \binom{4}{1} \times \binom{3}{1} \times 2^2 = 48 + 48 = 96.$

So, the correct answer is $96.$