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

Step 1: Locate the exact claim made by the passage about combinatorics and physical modeling. The passage says: "This kind of counting is sometimes used in the modeling of several physical phenomena." The word "sometimes" is a qualifier, not a universal claim.

Step 2: Evaluate statement P: "Combinatorics is always invoked in the modeling of physical phenomena." Since the passage explicitly uses "sometimes," it does not support an "always" claim. Using "always" instead of "sometimes" is a classic overgeneralization trap in reading comprehension. Hence, P is False.

Step 3: Evaluate statement Q: "Modeling some physical phenomena involves assigning probabilities to combinatorial possibilities in order to compute average values of physical quantities." Compare this directly with the passage: "the different combinatorial possibilities are assigned probability values. Assigning probabilities enables the computation of the average values of physical quantities." This is a faithful restatement, and the qualifier "some" correctly matches the passage's "sometimes." Hence, Q is True.

Step 4: Combine the two truth values: P is False, Q is True.

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

Final Answer: (B) P is False and Q is True