There are 4 white balls with the numbers $1, 2, 3, 4$ written on them and 3 black balls with the numbers $4, 5, 6$ written on them. When these 7 balls are randomly arranged in a line, the probability that the balls with the same number do not lie adjacent to each other is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

From the numbers $1,2,3,4,5,6$, five numbers are selected with repetition allowed to satisfy the following conditions, and then all five-digit natural numbers that can be formed by arranging them in a line are counted. What is the number of such natural numbers? [4 points] (가) Each odd number is either not selected or selected exactly once. (나) Each even number is either not selected or selected exactly twice.
(1) 450
(2) 445
(3) 440
(4) 435
(5) 430

From the numbers $1,2,3,4,5,6$, select five numbers with repetition allowed and arrange them in a line to form a five-digit natural number, satisfying the following conditions. How many such five-digit natural numbers can be formed? [4 points] (가) Each odd number is either not selected or selected exactly once. (나) Each even number is either not selected or selected exactly twice.

There are 5 cards with letters $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$ written on them and 4 cards with numbers $1,2,3,4$ written on them. When all 9 cards are arranged in a line in random order using each card once, what is the probability that the card with letter A has number cards on both sides? [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 12 }$

For the set $X = \{ 1,2,3,4 \}$, how many functions $f : X \rightarrow X$ satisfy the following condition? [3 points] $\square$
(1) 64
(2) 68
(3) 72
(4) 76
(5) 80

There are 6 students including students $\mathrm { A }$, $\mathrm { B }$, and $\mathrm { C }$. These 6 students sit around a circular table at equal intervals satisfying the following conditions. How many ways are there to seat them? (Note: arrangements that coincide by rotation are considered the same.) [4 points] (가) A and B are adjacent. (나) B and C are not adjacent.
(1) 32
(2) 34
(3) 36
(4) 38
(5) 40

There are 6 students including three students $\mathrm { A } , \mathrm { B } , \mathrm { C }$.
Find the number of ways these 6 students can all sit around a circular table with equal spacing satisfying the following conditions. (Note: rotations that coincide are considered the same.) [4 points] (가) A and B are adjacent. (나) B and C are not adjacent.

Among four-digit natural numbers that can be formed by selecting 4 numbers from the digits 1, 2, 3, 4, 5 with repetition allowed and arranging them in a line, how many are odd numbers greater than or equal to 4000? [3 points]
(1) 125
(2) 150
(3) 175
(4) 200
(5) 225

For the set $X = \{ x \mid x \text{ is a natural number not exceeding } 10 \}$, find the number of functions $f : X \rightarrow X$ satisfying the following conditions. [4 points] (가) For all natural numbers $x$ not exceeding 9, $f ( x ) \leq f ( x + 1 )$. (나) When $1 \leq x \leq 5$, $f ( x ) \leq x$, and when $6 \leq x \leq 10$, $f ( x ) \geq x$. (다) $f ( 6 ) = f ( 5 ) + 6$

The number of ways to arrange all 5 letters $x, x, y, y, z$ in a row is? [2 points]
(1) 10
(2) 20
(3) 30
(4) 40
(5) 50

Find the total number of ordered quadruples $(a, b, c, d)$ of natural numbers not exceeding 6 that satisfy the following condition. [4 points] $$a \leq c \leq d \text{ and } b \leq c \leq d$$

For the set $X = \{1, 2, 3, 4, 5, 6\}$, how many functions $f : X \rightarrow X$ satisfy the following conditions? [4 points] (가) The value of $f(1) \times f(6)$ is a divisor of 6. (나) $2f(1) \leq f(2) \leq f(3) \leq f(4) \leq f(5) \leq 2f(6)$
(1) 166
(2) 171
(3) 176
(4) 181
(5) 186

There are 10 empty bags arranged in a row, and 8 balls. Distribute the balls into the bags so that each bag contains at most 2 balls. Find the number of cases satisfying the following conditions. (Here, the balls are indistinguishable from each other.) [4 points] (가) The number of bags containing 1 ball is either 4 or 6. (나) Bags adjacent to a bag containing 2 balls contain no balls.

6. Using the digits $0, 1, 2, 3, 4, 5$ to form five-digit numbers with no repeated digits, the number of even numbers greater than $40000$ is
(A) $144$
(B) $120$
(C) $96$
(D) $72$

15. (13 points) Three table tennis associations have 27, 9, and 18 members respectively. Using stratified sampling, 6 athletes are selected from these three associations to participate in a competition. (I) Find the number of athletes to be selected from each of the three associations respectively; (II) The 6 selected athletes are numbered $A _ { 1 } , A _ { 2 } , A _ { 3 } , A _ { 4 } , A _ { 5 } , A _ { 6 }$ respectively. Two athletes are randomly selected from these 6 athletes to participate in a doubles match.
(i) List all possible outcomes using the given numbering;
(ii) Let event $A$ be ``at least one of the two athletes numbered $A _ { 5 }$ and $A _ { 6 }$ is selected''. Find the probability of event $A$ occurring.

17. (This question is worth 12 points)

A minibus has 5 seats numbered $1,2,3,4,5$. Passengers $P _ { 1 } , P _ { 2 } , P _ { 3 } , P _ { 4 } , P _ { 5 }$ have assigned seat numbers $1,2,3,4,5$ respectively. They board in order of increasing seat numbers. Passenger $P _ { 1 }$ did not sit in seat 1 due to health reasons. The driver then requires the remaining passengers to be seated according to the following rule: if their own seat is empty, they must sit in their own seat; if their own seat is occupied, they can choose any of the remaining empty seats. (1) If passenger $P _ { 1 }$ sits in seat 3 and other passengers are seated according to the rule, there are 4 possible seating arrangements. The table below shows two of them. Please fill in the remaining two arrangements (enter the seat numbers where passengers sit in the blank cells);
(2) If passenger

Passenger	$P _ { 1 }$	$P _ { 2 }$	$P _ { 3 }$	$P _ { 4 }$	$P _ { 5 }$
\multirow{3}{*}{Seat Number}	3	2	1	4	5
	3	2	4	5	1

$P _ { 1 }$ sits in seat 2, and other passengers are seated according to the rule, find the probability that passenger $P _ { 5 }$ sits in seat 5.

6. The number of ways to arrange 3 people in a row is
A. $12$ ways
B. $18$ ways
C. $24$ ways
D. $36$ ways [Figure] [Figure] [Figure] [Figure] [Figure]
C. C and D can know each other's scores
B. B can know all four people's scores

Chinese mathematician Chen Jingrun achieved world-leading results in research on Goldbach's conjecture. Goldbach's conjecture states that ``every even number greater than 2 can be expressed as the sum of two prime numbers'', such as $30 = 7 + 23$. Among prime numbers not exceeding 30, if two different numbers are randomly selected, the probability that their sum equals 30 is
A. $\frac { 1 } { 12 }$
B. $\frac { 1 } { 14 }$
C. $\frac { 1 } { 15 }$
D. $\frac { 1 } { 18 }$

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 ``- -''. The figure shows a hexagram. 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. $\frac { 11 } { 16 }$

Four students participate in garbage classification publicity activities in 3 residential areas, with each student going to only 1 area and each area having at least 1 student assigned. The total number of different arrangement methods is $\_\_\_\_$.

10. Three 1's and two 0's are randomly arranged in a row. The probability that the two 0's are not adjacent is
A. 0.3
B. 0.5
C. 0.6
D. 0.8

From 6 cards labeled $1,2,3,4,5,6$ respectively, 2 cards are randomly drawn without replacement. The probability that one of the drawn numbers is a multiple of the other is
A. $\frac { 1 } { 5 }$
B. $\frac { 1 } { 3 }$
C. $\frac { 2 } { 5 }$
D. $\frac { 2 } { 3 }$

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

Five volunteers participate in community service over Saturday and Sunday. Each day, two people are randomly selected from them to participate. The number of ways to select such that exactly one person participates on both days is
A. $120$
B. $60$
C. $40$
D. $30$

In the $4 \times 4$ grid table shown in the figure, select 4 squares such that each row and each column contains exactly one selected square. The total number of ways to do this is $\_\_\_\_$. Among all selections satisfying the above requirement, the maximum sum of the 4 numbers in the selected squares is $\_\_\_\_$.

11	21	31	40
12	22	33	42
13	22	33	43
15	24	34	44