Question

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?

• 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:

Watch for the shift from "sometimes" in the passage to "always" in statement P - that shift alone makes P false, while Q is a direct paraphrase of the passage and is true.

Answer 1

The Correct Option is B

This is an inference question, so the trick is to check whether each statement matches the strength of the passage's language, not just its topic.

Checking P: P claims combinatorics is always used in physical modeling. The passage only says counting is "sometimes" used - a hedge word. Any option that converts a "sometimes" claim into an "always" claim is a distortion, so P must be marked False.

Checking Q: Q says that in some models, probabilities are assigned to combinatorial possibilities specifically to compute averages of physical quantities. This lines up word-for-word with the passage's own description of what happens "often" in such models - combinatorial possibilities get probability values, and that lets you compute average values of physical quantities. So Q accurately reflects the passage and is True.

Putting it together: P is false (overclaim), Q is true (accurate restatement).

\(\text{Final: P False, Q True}\)

Correct option: (B)