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:

Look for words like "always" versus "sometimes" in the passage before accepting a strong claim like P.

Answer 1

The Correct Option is B

Step 1: Read the passage carefully for the exact wording.
The passage says this kind of counting is "sometimes used" in the modeling of physical phenomena, not "always used". It also says that often, in such models, combinatorial possibilities are assigned probability values, and this lets us compute average values of physical quantities.

Step 2: Check statement P.
P claims combinatorics is always invoked in the modeling of physical phenomena. The passage only says combinatorics is sometimes used for this purpose, not always. A single word like "sometimes" cannot support a claim of "always". So P is FALSE.

Step 3: Check statement Q.
Q claims that modeling some physical phenomena involves assigning probabilities to combinatorial possibilities in order to compute average values of physical quantities. This matches the passage almost word for word: the different combinatorial possibilities are assigned probability values, and assigning probabilities enables the computation of the average values of physical quantities. So Q is TRUE.

Step 4: Rule out the other options.
Since P is False and Q is True, the options that say P is True are wrong, and the option that says Q is False is also wrong.

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