14. From the subsets of the set $U = \{ a , b , c , d \}$, select 2 different subsets that must satisfy both of the following conditions: (1) Both $a$ and $b$ must be selected; (2) For any two selected subsets $A$ and $B$, we must have $A \subseteq B$ or $B \subseteq A$. Then there are $\_\_\_\_$ $36$ different ways.
Analysis: By enumeration, there are 36 ways in total.
II. Multiple Choice Questions (Total Score: 20 points) This section contains 4 questions. Each question has exactly one correct answer. Candidates must shade the box corresponding to the correct answer on the answer sheet. Each correct answer is worth 5 points; otherwise, zero points are awarded. 15. ``$x = 2 k \pi + \frac { \pi } { 4 } ( k \in \mathbb{Z} )$'' is a \_\_\_\_ condition for ``$\tan x = 1$''. [Answer] (A) (A) Sufficient but not necessary condition. (B) Necessary but not sufficient condition. (C) Sufficient condition. (D) Neither sufficient nor necessary condition. Analysis: $\tan \left( 2 k \pi + \frac { \pi } { 4 } \right) = \tan \frac { \pi } { 4 } = 1$, so it is sufficient; However, the converse does not hold. For example, $\tan \frac { 5 \pi } { 4 } = 1$, so it is not necessary.
(5 points) Let $y = f ( x )$ be a function whose graph is a continuous curve on the interval $(0,1]$, and $0 \leqslant f ( x ) \leqslant 1$ always holds. The area $S$ enclosed by the curve $y = f ( x )$ and the lines $x = 0 , x = 1 , y = 0$ can be estimated using the Monte Carlo method. First, generate two groups of $N$ uniformly distributed random numbers on the interval $(0,1]$: $x _ { 1 } , x _ { 2 } , \ldots , x _ { N }$ and $y _ { 1 } , y _ { 2 } , \ldots , y _ { N }$, obtaining $N$ points $( x_i , y_i )$ $(i = 1,2 \ldots , N)$. Then count the number $N _ { 1 }$ of points satisfying $y _ { i } \leqslant f ( x_i )$ $(i = 1,2 \ldots , N)$. By the Monte Carlo method, the approximate value of $S$ is $\_\_\_\_$.
14. From the subsets of the set $U = \{ a , b , c , d \}$, select 2 different subsets that must satisfy both of the following conditions:\\
(1) Both $a$ and $b$ must be selected;\\
(2) For any two selected subsets $A$ and $B$, we must have $A \subseteq B$ or $B \subseteq A$. Then there are $\_\_\_\_$ $36$ different ways.
\section*{Analysis: By enumeration, there are 36 ways in total.}
II. Multiple Choice Questions (Total Score: 20 points) This section contains 4 questions. Each question has exactly one correct answer. Candidates must shade the box corresponding to the correct answer on the answer sheet. Each correct answer is worth 5 points; otherwise, zero points are awarded.
15. ``$x = 2 k \pi + \frac { \pi } { 4 } ( k \in \mathbb{Z} )$'' is a \_\_\_\_ condition for ``$\tan x = 1$''.\\
[Answer] (A)\\
(A) Sufficient but not necessary condition.\\
(B) Necessary but not sufficient condition.\\
(C) Sufficient condition.\\
(D) Neither sufficient nor necessary condition.
Analysis: $\tan \left( 2 k \pi + \frac { \pi } { 4 } \right) = \tan \frac { \pi } { 4 } = 1$, so it is sufficient;\\
However, the converse does not hold. For example, $\tan \frac { 5 \pi } { 4 } = 1$, so it is not necessary.\\