QUESTION : Design F $=(1,2,5,7,8,10,15)$ using only one of the options below:

a$.8-$to$-1$ MUX to get the full marks.

$($the image is the answer$)$

I$$ve seen the given question and the answer you$$ve provided for designing F $=(1,2,5,7,8,10,15)$ using an $8-$to$-1$ MUX, but I$$m having a hard time understanding the reasoning behind the final arrangement. Could you please break down the steps, like taking lecture$-$style notes, to explain how and why the solution was obtained?

Specifically, I want to understand:

How the original function$$s minterms translate to which inputs of the MUX should be high or low.

Why certain input lines of the MUX correspond to certain minterm indices.

How the select lines $($W$,$ Z$,$ Y$)$ are chosen and how the mapping from W$,$ Z$,$ Y to the inputs relates to the minterm numbers.

The logical process or $$algorithm$$ for deciding what goes where on the MUX inputs and why that gives the correct output for all combinations.

In short, I$$m not just looking for the final wiring diagram. I$$d like a clear, step$-$by$-$step explanation of the thought process behind it$,$ written as if it were organized classroom notes. Thanks!