A logic problem involving conjunctions and disjunctions 
*

*$(P \vee Q)$ & $R$ 

*$(R$ & $P) \rightarrow S$

*$(Q$ & $R) \rightarrow S$ 


Conclusion is $S$
without using conditional proof, please?
And perhaps if you have time, please show me the conditional proof method or at least which CP you would use.
Thank you & I do appreciate your time.
My difficulty is that I've been learning from a book, that does not actually give all the answers to the problems set.
The ones not given are seemingly more difficult than the answered ones.
I did try


*$Q~~~~~~$    2$~~$ SIMP

*$P$&$Q~$  1,4 CONJ

*$R~~~~~~$    1$~~~$ SIMP


I did see/thought I did a Constructive Dilemma CD but was not a 100% certain I could apply it.
Apologies for not showing that I had actually made an effort.
:D
Critical thinking 9th Edition by Richard Parker & Brooke Noel Moore, p. 339 Q2.
 A: \begin{align*}
\text{Given:}&&1.&\;\; (P \lor Q) \land R\\[2pt]
&&2.&\;\; (R \land P) \to S\\[2pt]
&&3.&\;\;(Q \land R) \to S\\[12pt]
\text{To prove:}&&S&\\[12pt]
\text{Proof:}&&4.&\;\; P \lor Q&&\text{[$1$, SIM]}\\[2pt]
&&5.&\;\; R&&\text{[$1$, SIM]}\\[12pt]
&&6.&\;\; \lnot (R \land P) \lor S&&\text{[$2$, IMP]}\\[2pt]
&&7.&\;\; (\lnot R \lor \lnot P) \lor S&&\text{[$6$, DEM]}\\[2pt]
&&8.&\;\; \lnot R \lor (\lnot P \lor S)&&\text{[$7$, ASSOC]}\\[2pt]
&&9.&\;\; R \to (\lnot P \lor S)&&\text{[$8$, IMP]}\\[2pt]
&&10.&\;\; \lnot P \lor S&&\text{[$9,5$, MP]}\\[12pt]
&&11.&\;\;(R \land Q) \to S&&\text{[$3$, COM]}\\[2pt]
&&12.&\;\; \lnot (R \land Q) \lor S&&\text{[$11$, IMP]}\\[2pt]
&&13.&\;\; (\lnot R \lor \lnot Q) \lor S&&\text{[$12$, DEM]}\\[2pt]
&&14.&\;\; \lnot R \lor (\lnot Q \lor S)&&\text{[$13$, ASSOC]}\\[2pt]
&&15.&\;\; R \to (\lnot Q \lor S)&&\text{[$14$, IMP]}\\[2pt]
&&16.&\;\; \lnot Q \lor S&&\text{[$15,5$, MP]}\\[12pt]
&&17.&\;\; (\lnot P \lor S) \land (\lnot Q \lor S) &&\text{[$10,16$, CONJ]}\\[2pt]
&&18.&\;\; (\lnot P \land \lnot Q) \lor S &&\text{[$17$, DIST]}\\[2pt]
&&19.&\;\; \lnot (P \lor Q) \lor S &&\text{[$18$, DEM]}\\[2pt]
&&20.&\;\; (P \lor Q) \to S &&\text{[$19$, IMP]}\\[2pt]
&&21.&\;\; S &&\text{[$20,4$, MP]}\\[2pt]
\end{align*}
A: This version is almost the same as my first answer, and it's the same length, but by making use of "Constructive Dilemma", it's perhaps conceptually a little simpler. The proofs are exactly the same for the first $10$ steps.
\begin{align*}
\text{Given:}&&1.&\;\; (P \lor Q) \land R\\[2pt]
&&2.&\;\; (R \land P) \to S\\[2pt]
&&3.&\;\;(Q \land R) \to S\\[12pt]
\text{To prove:}&&S&\\[12pt]
\text{Proof:}&&4.&\;\; P \lor Q&&\text{[$1$, SIM]}\\[2pt]
&&5.&\;\; R&&\text{[$1$, SIM]}\\[12pt]
&&6.&\;\; \lnot (R \land P) \lor S&&\text{[$2$, IMP]}\\[2pt]
&&7.&\;\; (\lnot R \lor \lnot P) \lor S&&\text{[$6$, DEM]}\\[2pt]
&&8.&\;\; \lnot R \lor (\lnot P \lor S)&&\text{[$7$, ASSOC]}\\[2pt]
&&9.&\;\; R \to (\lnot P \lor S)&&\text{[$8$, IMP]}\\[2pt]
&&10.&\;\; \lnot P \lor S&&\text{[$9,5$, MP]}\\[2pt]
&&11.&\;\; P \to S&&\text{[$10$, IMP]}\\[12pt]
&&12.&\;\;(R \land Q) \to S&&\text{[$3$, COM]}\\[2pt]
&&13.&\;\; \lnot (R \land Q) \lor S&&\text{[$12$, IMP]}\\[2pt]
&&14.&\;\; (\lnot R \lor \lnot Q) \lor S&&\text{[$13$, DEM]}\\[2pt]
&&15.&\;\; \lnot R \lor (\lnot Q \lor S)&&\text{[$14$, ASSOC]}\\[2pt]
&&16.&\;\; R \to (\lnot Q \lor S)&&\text{[$15$, IMP]}\\[2pt]
&&17.&\;\; \lnot Q \lor S&&\text{[$16,5$, MP]}\\[2pt]
&&18.&\;\; Q \to S&&\text{[$17$, IMP]}\\[12pt]
&&19.&\;\; (P \lor Q) \to (S \lor S)&&\text{[$11,18,4$, CD]}\\[2pt]
&&20.&\;\; S \lor S&&\text{[$19,4$, MP]}\\[2pt]
&&21.&\;\; S &&\text{[$20$, TAUT]}\\[2pt]
\end{align*}
