Question

Combinatorics deals with problems involving counting... 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?

• A. P is False and Q is False.
• B. P is False and Q is True.
• C. P is True and Q is False.
• D. P is True and Q is True.

Hint:

Compare the quantifier words: the passage says "sometimes"/"often", not "always".

Answer 1

The Correct Option is B

This is a critical reading question that tests whether the reader distinguishes between absolute claims ("always") and qualified claims ("sometimes", "often", "some") in a passage, and whether a given inference is directly supported by the text or overreaches it.

1. Evaluating statement P: P claims "Combinatorics is always invoked in the modeling of physical phenomena." The passage states that this kind of counting is "sometimes used in the modeling of several physical phenomena" - the word "sometimes" explicitly signals that combinatorics is used only in some cases of modeling physical phenomena, not in every case. Since the passage never asserts universality, and in fact uses a word that directly contradicts universality, statement P goes beyond what the passage supports. Therefore P is False.
2. Evaluating statement Q: Q claims "Modeling some physical phenomena involves assigning probabilities to combinatorial possibilities in order to compute average values of physical quantities." The passage states: "Often, in such models, the different combinatorial possibilities are assigned probability values. Assigning probabilities enables the computation of the average values of physical quantities." This is precisely the chain described in Q - combinatorial possibilities get probability values, and this assignment enables computing average values. Because Q only claims this happens for "some" phenomena (a modest, qualified claim, consistent with the passage's own "sometimes"/"often" language), and the passage directly describes this exact mechanism, Q is fully supported by the text. Therefore Q is True.
3. Why the other options are wrong: Option (A) "P is False and Q is False" is wrong because Q is in fact supported by the passage, not false. Option (C) "P is True and Q is False" is wrong on both counts: P overstates the passage ("always" versus "sometimes"), and Q is actually true, not false. Option (D) "P is True and Q is True" is wrong because P's claim of universality ("always") is directly contradicted by the passage's own wording ("sometimes").

Since P overreaches the passage's qualified language and Q accurately reflects it, the correct inference is that P is False and Q is True.

\[ \boxed{\text{P is False and Q is True}} \]