Utilizing the expansion theorem, express the following as symmetric functions:

a$.$ A$\text{'}$S$0,1,4($B$,$C$,$D$,$E$)+$ AS$\text{'}0,3,4($B$,$C$,$D$,$E$)$

b$.$ A$\text{'}$S$0,1\mathrm{.}4($B$,$C$,$D$,$E$)+$ AS$0,3,4($B$,$C$,$D$,$E$)$

c$.$ A$\text{'}$S$0,1,4($B$,$C$,$D$,$E$)+$ AS$\text{'}0,3,4($B$\text{'},$ C$\text{'},$D$\text{'},$E$\text{'})$

$7.$ For each of the following functions, find a contact$-$network realization with minimum number of contacts.

a$.$ S$1,4($w$,$x$,$y$,$z$)$

b$.$ S$0,1,3($w$,$x$,$y$,$z$)$