A Boolean digital circuit is composed using two 4-input multiplexers (M1 and M2) and one 2-input multiplexer (M3) as shown in the figure. X0-X7 are the inputs of the multiplexers M1 and M2 and could be connected to either 0 or 1. The select lines of the multiplexers are connected to Boolean variables A, B and C as shown.

Which one of the following set of values of (X0, X1, X2, X3, X4, X5, X6, X7) will realise the Boolean function $$\overline A + \overline A \,.\,\overline C + A\,.\,\overline B \,.\,C$$ ?

Consider the two cascaded $$2$$-to-$$1$$ multiplexers as shown in the figure.

The minimal sum of products form of the output $$X$$ is

Consider the $$4$$-to-$$1$$ multiplexer with two select lines $${S_1}$$ and $${S_0}$$ given below

The minimal sum-of-products form of the Boolean expression for the output $$F$$ of the multiplexer is

Suppose only one multiplexer and one inverter are allowed to be used to implement any Boolean function of $$n$$ variables. What is the minimum size of the multiplexer needed?