Step 1: Understand the two routing families.
Flood routing predicts how a flood wave changes as it moves through a reservoir or along a channel.
There are two broad families: hydrologic routing, which uses the continuity equation together with a storage-discharge relation, and hydraulic routing, which solves the full St. Venant (continuity + momentum) equations for the flow.
Reservoir routing and channel routing are both special cases of hydrologic routing, so a hydrologic-routing method can correctly appear against either of them, or against the general "hydrologic routing" heading itself.
Step 2: Match each method to its correct routing type.
The Muskingum method models storage in a channel reach as a combination of prism storage and wedge storage, and it is the classic hydrologic method used for channel routing. So (S) Channel routing - (i) Muskingum method.
The Modified Puls method (also called the storage-indication method) routes a flood through a reservoir using the storage-outflow relation, so (R) Reservoir routing - (ii) Modified Pul's method.
The Method of Characteristics solves the St. Venant equations directly along characteristic curves in space-time, which is the defining feature of hydraulic routing, so (Q) Hydraulic routing - (iii) Method of Characteristics.
Since (P) Hydrologic routing is the parent category of both reservoir routing and channel routing, it is correctly matched with EITHER (i) Muskingum method or (ii) Modified Pul's method, because both are hydrologic methods.
Step 3: Check option (A): (P)-(i); (Q)-(iii); (R)-(ii).
(P) Hydrologic routing paired with (i) Muskingum method is valid, since Muskingum is a hydrologic method.
(Q) Hydraulic routing paired with (iii) Method of Characteristics is exactly correct.
(R) Reservoir routing paired with (ii) Modified Pul's method is exactly correct.
All three pairs in option (A) are true, so (A) is a CORRECT option.
Step 4: Check option (B): (Q)-(i); (R)-(iii); (S)-(ii).
(Q) Hydraulic routing paired with (i) Muskingum method is wrong, because Muskingum is a hydrologic method, not a hydraulic one.
(R) Reservoir routing paired with (iii) Method of Characteristics is wrong, because reservoir routing is done by the storage-indication (Modified Puls) method, not by solving the St. Venant equations.
(S) Channel routing paired with (ii) Modified Pul's method is wrong, because channel routing by the hydrologic approach uses the Muskingum method.
Every pair in option (B) is false, so (B) is NOT a correct option.
Step 5: Check option (C): (P)-(i); (R)-(ii); (S)-(i).
(P) Hydrologic routing with (i) Muskingum method is valid, as in Step 3.
(R) Reservoir routing with (ii) Modified Pul's method is exactly correct.
(S) Channel routing with (i) Muskingum method is exactly correct.
All three pairs hold, so (C) is a CORRECT option.
Step 6: Check option (D): (P)-(ii); (Q)-(iii); (R)-(ii).
(P) Hydrologic routing with (ii) Modified Pul's method is valid, because Modified Puls is also a hydrologic method (used for reservoirs, a sub-type of hydrologic routing).
(Q) Hydraulic routing with (iii) Method of Characteristics is exactly correct.
(R) Reservoir routing with (ii) Modified Pul's method is exactly correct.
All three pairs hold, so (D) is a CORRECT option.
Final Answer:
Options (A), (C) and (D) give correct matches; option (B) does not.
\[ \boxed{(A), (C), (D)} \]