The number of ways to distribute 8 identical chocolates to four students $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ according to the following rules is? [3 points] (가) Each student receives at least 1 chocolate. (나) Student A receives more chocolates than student B.
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19

A bag contains 3 white balls and 4 black balls. When drawing 4 balls simultaneously at random from the bag, what is the probability of drawing 2 white balls and 2 black balls? [3 points]
(1) $\frac { 2 } { 5 }$
(2) $\frac { 16 } { 35 }$
(3) $\frac { 18 } { 35 }$
(4) $\frac { 4 } { 7 }$
(5) $\frac { 22 } { 35 }$

How many ordered pairs $( a , b , c , d )$ of non-negative integers satisfy the following conditions? [4 points] (가) $a + b + c - d = 9$ (나) $d \leq 4$ and $c \geq d$
(1) 265
(2) 270
(3) 275
(4) 280
(5) 285

Find the value of ${ } _ { 7 } \mathrm { P } _ { 2 } + { } _ { 7 } \mathrm { C } _ { 2 }$. [3 points]

Three students A, B, and C are given 6 identical candies and 5 identical chocolates to be distributed completely according to the following rules. Find the number of ways to do this. [4 points] (가) The number of candies that student A receives is at least 1. (나) The number of chocolates that student B receives is at least 1. (다) The sum of the number of candies and chocolates that student C receives is at least 1.

Find the number of ways to distribute 6 black hats and 6 white hats among four students $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ according to the following rules without remainder. (Note: hats of the same color are not distinguished from each other.) [4 points] (가) Each student receives at least 1 hat. (나) The number of black hats each student receives is different from one another.

How many ordered pairs $( a , b , c , d , e )$ of natural numbers satisfy the following conditions? [3 points]
(a) $a + b + c + d + e = 12$
(b) $\left| a ^ { 2 } - b ^ { 2 } \right| = 5$
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

For two sets $X = \{ 1,2,3,4,5 \} , Y = \{ 1,2,3,4 \}$, how many functions $f$ from $X$ to $Y$ satisfy the following conditions? [4 points]
(a) For all elements $x$ in set $X$, $f ( x ) \geq \sqrt { x }$.
(b) The range of function $f$ has exactly 3 elements.
(1) 128
(2) 138
(3) 148
(4) 158
(5) 168

A survey was conducted on the preferences for Subject A and Subject B among 16 students in a class. Each student who participated in the survey chose one of the two subjects. 9 students chose Subject A and 7 students chose Subject B. When 3 students are randomly selected from the 16 students who participated in the survey, what is the probability that at least one of the 3 selected students chose Subject B? [3 points]
(1) $\frac{3}{4}$
(2) $\frac{4}{5}$
(3) $\frac{17}{20}$
(4) $\frac{9}{10}$
(5) $\frac{19}{20}$

How many ways are there to select 3 letters from the four letters $a , b , c , d$ with repetition allowed and arrange them in a row? [2 points]
(1) 56
(2) 60
(3) 64
(4) 68
(5) 72

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.

9. Let $A = \{(x,y) \mid x^2 + y^2 \leq 1, x, y \in \mathbf{Z}\}$, $B = \{(x,y) \mid |x| \leq 2, |y| \leq 2, x, y \in \mathbf{Z}\}$. Define $A \oplus B = \{(x_1 + x_2, y_1 + y_2) \mid (x_1, y_1) \in A, (x_2, y_2) \in B\}$. The number of elements in $A \oplus B$ is
A. 77
B. 49
C. 45
D. 30

17. Two schools A and B organize student teams to participate in a debate competition. School A recommends $3$ male students and $2$ female students, while school B recommends $3$ male students and $4$ female students. The students recommended by both schools participate in training. Since the students' levels are comparable after training, $3$ people are randomly selected from the male students and $3$ people are randomly selected from the female students to form a representative team.
(1) Find the probability that at least $1$ student from school A is selected for the representative team.
(2) Before a certain competition, $4$ people are randomly selected from the $6$ team members to participate. Let $X$ denote the number of male students participating, find the probability distribution and mathematical expectation of $X$.

From 2 male students and 3 female students, 2 people are selected to participate in community service. The probability that both selected are female students is
A. 0.6
B. 0.5
C. 0.4
D. 0.3

From 2 female students and 4 male students, select 3 people to participate in a science and technology competition, with at least 1 female student selected. The total number of different selection methods is $\_\_\_\_$ (Answer with numerals)

6. In ancient Chinese classics, the ``Book of Changes'' uses ``hexagrams'' to describe the changes of all things. Each ``hexagram'' consists of 6 lines arranged from bottom to top, with lines divided into yang lines ``——'' and yin lines ``——'', as shown in the figure. If a hexagram is randomly selected from all hexagrams, the probability that it has exactly 3 yang lines is
A. $\frac { 5 } { 16 }$
B. $\frac { 11 } { 32 }$
C. $\frac { 21 } { 32 }$
D. $\frac { 11 } { 16 }$

From 6 people, select 4 to work on duty, each person works for 1 day. The first day needs 1 person, the second day needs 1 person, the third day needs 2 people. There are $\_\_\_\_$ ways to arrange them.

10. Arrange 4 ones and 2 zeros randomly in a row. The probability that the 2 zeros are not adjacent is [Figure]
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{2}{3}$
D. $\frac{4}{5}$

From 5 classmates including A and B, 3 are randomly selected to participate in community service work. The probability that both A and B are selected is $\_\_\_\_$.

If 4 vertices are randomly selected from the 8 vertices of a cube, the probability that these 4 points lie on the same plane is $\_\_\_\_$

An urn contains three green dice and two red dice. Two dice are randomly drawn from the urn in one draw. Give a term with which one can determine the probability that one red die and one green die are drawn.

We have $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ where $D_n = \left\{\sum_{j=1}^{n} \frac{x_j}{2^j}, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n\right\}$.
Is the set $D$ countable?

104- From each of 6 regions of the country, 15 students are invited to a cultural center. In how many ways can 3 students be selected from among them such that no two of them are from the same region?

• [(1)] $\Delta7600$
• [(2)] $67\Delta00$
• [(3)] $7\Delta600$
• [(4)] $76\Delta00$

146- If $A = \{2k-1 \mid k \in \mathbb{Z},\, 1 \leq k \leq 5\}$ and $B = \{k \in \mathbb{Z} : |k-3| \leq 2\}$, then the set $(A \times B) \cap (B \times A)$ has how many elements?
(1) $6$ (2) $8$ (3) $9$ (4) $16$
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153. In how many ways can 9 identical books be placed in 5 shelves such that at least one book is placed on each shelf?
(1) $35$ (2) $42$ (3) $56$ (4) $70$