To implement a ful adder using $4\times 1$ multiplexer first we will draw the truth table of a full adder.

 We are getting two outputs Sum (S) and Carry (Cout).

The boolean function of S and Cout can be written as,

$S=\Sigma m (1,2,4,7)\\[2ex]C_{out}=\Sigma m(3,5,6,7)$

Since $4\times 1$ mux is to be used, it will have only one input and two selection lines.

Here we use $A$ as an input nd $B, C$ as selection lines. The two $4\times 1$ mux are cascaded. * The input$A$ and$\bar{A}$ are connected to lines I0, I1, I2 and I3 respectively of$4\times 1$ mux for sum. * The inpuut logic '0',$A$,$\bar{A}$ and logic '1' are connected to inputs of I4, I5, I6 and I7 respectively of$4\times 1$ mux for Cout.