Someome solve this plzz... This is from Mathematical logic

Someome solve this plzz... This is from Mathematical logic 27. Give alternative arrangement of the following circuit. so that the new circuit has minimum switches only

let 
p : The switch S₁ is closed.
q : The switch S2 is closed.
r : The switch S₃ is closed.
~p : The switch S₁` is closed.
~q : The switch S<sub>2</sub>` is closed.
The symbolic fom of the given cir circuit is :
Note : PLEASE CONSIDER 'A' AS AND'S SIGN AND 'V' AS OR 'S SIGN.
(p A q A ~p) V (~p A q A r) V (p A q A r) V (p A ~q A r)

= (p A ~p A q) V (~p A q A r) V (p A q A r) V (p A ~q A r)     - By Commutative Law
= (F A q) V (~p A q A r) V (p A q A r) V (p A ~q A r)             - By Complement Law
= F V (~p A q A r) V (p A q A r) V (p A ~q A r)                      - By Identity Law
= (~p A q A r) V (p A q A r) V (p A ~q A r)                             - By Identity Law  
= {[~p A (q A r)] V [p A (q A r)]} V (p A ~q A r)                       - By Associative Law
= [(~p V p) A (q A r)] V (p A ~q A r)                                       - By Distributive Law
= [T A(q A r)] V (p A ~q A r)                                                   - By Complement Law
= (q A r) V (p A ~q A r)                                                           - By Identity Law
= [q V (p A ~q)] A r                                                                 - By Distributive Law
= [(q V p) A (q V ~q)] A r                                                        - By Distributive Law
= [(q V p) A T] A r                                                                   - By Complement Law
= (q V p) A r                                                                            - By Identity Law
=(p V q) A r                                                                             - By Commutative Law
I CONSIDER THAT YOU KNOW HOW TO DRAW CIRCUIT DIAGRAM.