Question

Consider a digital logic circuit consisting of three 2-to-1 multiplexers \(M_1, M_2,\) and \(M_3\) as shown below. \(X_1\) and \(X_2\) are inputs of \(M_1\). \(X_3\) and \(X_4\) are inputs of \(M_2\). \(A, B,\) and \(C\) are select lines of \(M_1, M_2,\) and \(M_3,\) respectively. \begin{center} \includegraphics[width=6cm]{64.png} \end{center} For an instance of inputs \(X_1 = 1\), \(X_2 = 1\), \(X_3 = 0\), and \(X_4 = 0\), the number of combinations of \(A, B, C\) that give the output \(Y = 1\) is \_\_\_\_\_.

Answer 1

\(M_1\) is controlled by select line \(A\):
If \(A = 0\), \(M_1\) selects \(X_1 = 1\).
If \(A = 1\), \(M_1\) selects \(X_2 = 1\).
In both cases, the output of \(M_1\) is \(1\).
\(M_2\) is controlled by select line \(B\):
If \(B = 0\), \(M_2\) selects \(X_3 = 0\).
If \(B = 1\), \(M_2\) selects \(X_4 = 0\).
In both cases, the output of \(M_2\) is \(0\).
\(M_3\) is controlled by select line \(C\):
If \(C = 0\), \(M_3\) selects the output of \(M_1 = 1\).
If \(C = 1\), \(M_3\) selects the output of \(M_2 = 0\).
For \(Y = 1\), \(C = 0\) is required.
The number of valid combinations of \(A, B, C\) is determined as follows:
\(A\) can take either \(0\) or \(1\) (2 possibilities).
\(B\) can take either \(0\) or \(1\) (2 possibilities).
\(C\) must be \(0\) (1 possibility).
Total combinations = \(2 \times 2 \times 1 = 4\). Final Answer: \[ \boxed{4} \]