Step-by-step Solution:

The number of elements in the power set ( P(S) ) of the set ( S = \(\emptyset\), 1, 2, 3 ) is:\[\] The power set of a set ( S ) is the set of all subsets of ( S ). The number of elements in the power set of a finite set with ( n ) elements is given by the formula ( 2^n ).\[\] Step 1: Count the number of elements in ( S ) The given set is ( S = \(\emptyset\), 1, 2,3 ). The elements of ( S ) are:\[\] 1. ( \(\emptyset\) ) (the empty set) \[\] 2. ( 1 ) \[\] 3. ( 2, 3 ) \[\] The number of elements in ( S ) is ( n(S) = 3 ).\[\] Step 2: Find the number of elements in ( P(S) ) The formula to calculate the number of elements in a power set is ( 2^n ), where ( n ) is the number of elements in the original set.\[\] Substitute ( n = 3 ): |P(S)| = \(2^3\) = 8 \[\] \[\] \[\] Correct Option: (d)