'When the teacher is in the room, all students stand silently.'

If the above statement is true, which one of the following statements is not necessarily true?

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 $Q$ ?

In Panel I of the figure below, the front view and top view of a structure are shown. Which one of the 3D structures shown in Panel II possesses the views shown in Panel I?

For positive real numbers $S$ and $K$, the function $H_K(S)$ is defined as: $H_K(S)=\max (S-K, 0)$. The max function is defined as:

$$ \max (a, b)= \begin{cases}a, & \text { when } a>b \\ b, & \text { when } a \leq b\end{cases} $$

The graph below shows the plot of a function $N(S)$ versus $S$. $N(S)$ can be expressed as $\_\_\_\_$ .