Consider a digital logic circuit consisting of three 2-to-1 multiplexers M1, M2, and M3 as shown below. X1 and X2 are inputs of M1. X3 and X4 are inputs of M2. A, B, and C are select lines of M1, M2, and M3, respectively. For an instance of inputs $X1=1$, $X2=1$, $X3=0$, and $X4=0$, the number of combinations of A, B, C that give the output $Y=1$ is _______

Given the inputs to the multiplexers:

• For multiplexer M1 with select line $A$ and inputs $X1 = 1, X2 = 1$:

  $$Q1 = \overline{A} \cdot X1 + A \cdot X2 = \overline{A}(1) + A(1) = 1$$

  $Q1 = 1$ irrespective of the value of $A$.

• For multiplexer M2 with select line $B$ and inputs $X3 = 0, X4 = 0$:

  $$Q2 = \overline{B} \cdot X3 + B \cdot X4 = \overline{B}(0) + B(0) = 0$$

  $Q2 = 0$ irrespective of the value of $B$.

• Multiplexer M3 has inputs $Q1$ (at line 0) and $Q2$ (at line 1), with select line $C$:

  $$Y = \overline{C} \cdot Q1 + C \cdot Q2 = \overline{C}(1) + C(0) = \overline{C}$$

For the output $Y = 1$, we must have:

Since $Y = 1$ only requires $C = 0$:

• $A$ can be $0$ or $1$ (2 possibilities)

• $B$ can be $0$ or $1$ (2 possibilities)

• $C$ must be $0$ (1 possibility)

Thus, the total number of combinations of $(A, B, C)$ is: