Combinatorics deals with problems involving counting. For example, "How many distinct arrangements of N distinct objects in M spaces on a circle are possible?" is a typical problem in combinatorics. This kind of counting is sometimes used in the modeling of several physical phenomena. Often, in such models, the different combinatorial possibilities are assigned probability values. Assigning probabilities enables the computation of the average values of physical quantities.
Consider the following statements:
P: Combinatorics is always invoked in the modeling of physical phenomena.
Q: Modeling some physical phenomena involves assigning probabilities to combinatorial possibilities in order to compute average values of physical quantities.
Based on the passage above, what can be inferred about statements P and Q?
Show Hint
Look for words like "always" versus "sometimes" in the passage before accepting a strong claim like P.
Step 1: Separate what the passage states as fact from what the statements claim.
The passage gives two pieces of information: combinatorics is "sometimes" used to model physical phenomena, and in such models probabilities are often assigned to combinatorial outcomes so that average values of physical quantities can be worked out.
Step 2: Test statement Q against the passage.
Q says modeling some physical phenomena uses probabilities on combinatorial possibilities to get average values. This is exactly what the passage describes in its last line. Nothing in Q goes beyond what the passage supports. So Q is TRUE.
Step 3: Test statement P against the passage.
P says combinatorics is always invoked in modeling physical phenomena. The word used in the passage is "sometimes", which is a weaker claim than "always". An "always" claim needs every case to hold, and the passage never says that. So P is FALSE.
Step 4: Match with the answer choices.
We need P false and Q true. Checking the four choices, only one option pairs a false P with a true Q, and the rest either make P true or make Q false, both of which go against what we found.
Step 5: State the result.
P is false because the passage only claims sometimes, not always. Q is true because it restates the passage directly.
\[ \boxed{\text{P is False and Q is True}} \]
