$3\mathrm{.}5$ A bank vault has three locks with a different key for each lock. Each key is owned by a different person. To open the door, at least two people must insert their keys into the assigned locks. Each input signal line $A,B,$ and $C$ is $1$ iff the correct key is inserted into lock $1,2,$ and $3,$ respectively. The output signal $Z$ is $1$ iff the door should open.

a$.$ Construct a truth table for the output $Z$ in terms of the input signals $A,B,$ and $C.$

b$.$ Find the minterm expansion for $Z$ from the truth table. Write the minterm expansion both in "normal" notation $($showing variables$/$literals$)$ and in $m-$notation.

c$.$ Find the maxterm expansion for $Z$ from the truth table. Write the maxterm expansion both in "normal" notation $($showing variables$/$literals$)$ and in $M-$notation.