The amount of ROM needed to implement a $4-bit$ multiplier is
A ROM cannot be written. So, to implement a $4$-bit multiplier we must store all the possible combinations of $2^4 \times 2^4$ inputs and their corresponding $8$ output bits giving a total of $ 2^4 \times 2^4 \times 8$ bits $= 2048$ bits. So, (D) is the answer.
PS: We are not storing the input bits explicitly -- those are considered in order while accessing the output $8$ bits. In this way, by storing all the possible outputs in order we can avoid storing the input combinations.
this image clarifies the ROM thing here..