Question

Consider the digital circuit shown below with two input lines A and B, two select
lines S0 and S1, and an output line Y. The blocks Q and M represent active high
2:4 decoder and 4-to-1 multiplexer, respectively. Out of 16 possible input
combinations, the number of combinations that produce Y=1 is ____________.
(answer in integer)
Note: One input combination is an instance of [A B S1 S0].

Hint:

Trace which decoder output line feeds each multiplexer data input, note any branched wire that ties one decoder output into two adjacent multiplexer inputs, then for every select value S1S0 count how many (A, B) pairs make the corresponding data input HIGH.

Answer 1

Correct Answer: 6

The circuit has a 2-to-4 active-high decoder \(Q\) whose two inputs are \(A\) and \(B\), and a 4-to-1 multiplexer \(M\) whose two select lines are \(S_1\) and \(S_0\). The four decoder outputs feed the four data lines of the multiplexer, and tracing the wires in the figure shows that some decoder output lines branch and feed two adjacent multiplexer data lines, so \(Y\) is not simply the decoder output picked by \(S_1S_0\).

Step 1: Decoder behaviour. For a 2:4 active-high decoder with address inputs \(A\) (MSB) and \(B\) (LSB), exactly one output is HIGH for every \((A,B)\) pair: \(D_0=1\) only for \(AB=00\), \(D_1=1\) only for \(AB=01\), \(D_2=1\) only for \(AB=10\), \(D_3=1\) only for \(AB=11\).

Step 2: Multiplexer data-line connections read from the figure: \(I_0=D_0\), \(I_1=D_1+D_2\), \(I_2=D_2+D_3\), \(I_3=D_3\). The multiplexer output is \(Y=I_k\), where \(k\) is the decimal value of \(S_1S_0\).

Step 3: Check each of the 4 possible \(S_1S_0\) values against the 4 possible \((A,B)\) values.

• \(S_1S_0=00 \Rightarrow Y=D_0\), true only for \(AB=00\): 1 combination.
• \(S_1S_0=01 \Rightarrow Y=D_1+D_2\), true for \(AB=01\) or \(AB=10\): 2 combinations.
• \(S_1S_0=10 \Rightarrow Y=D_2+D_3\), true for \(AB=10\) or \(AB=11\): 2 combinations.
• \(S_1S_0=11 \Rightarrow Y=D_3\), true only for \(AB=11\): 1 combination.

Step 4: Total count. Adding the counts for each select value gives \(1+2+2+1=6\). The 6 combinations of \([A\ B\ S_1\ S_0]\) are 0000, 0101, 1001, 1010, 1110 and 1111.

Final Answer: 6