There are six students (three female and three male) who frequently interact with a teacher at school. After graduation, the teacher invites them to a gathering. After the meal, seven people stand in a row for a commemorative photo. It is known that among the students, one female and one male had an unpleasant experience and do not want to stand adjacent during the photo, while the teacher stands in the middle and the three male students do not all stand on the same side of the teacher. The total number of possible arrangements is (17--1)(17--2)(17--3).

Arrange the digits $1, 2, 3, \ldots, 9$ into a nine-digit number (digits cannot be repeated) such that the first 5 digits are increasing from left to right and the last 5 digits are decreasing from left to right. How many nine-digit numbers satisfy the conditions?
(1) $\frac{8!}{4!4!}$
(2) $\frac{8!}{5!3!}$
(3) $\frac{9!}{5!4!}$
(4) $\frac{8!}{5!}$
(5) $\frac{9!}{5!}$

A department store holds a Father's Day card drawing promotion with the following rules: The organizer prepares ten cards numbered $1, 2, \ldots, 9$, of which there are two cards numbered 8, and only one card for each other number. Four cards are randomly drawn from these ten cards without replacement and arranged from left to right in order to form a four-digit number. If the four-digit number satisfies any one of the following conditions, a prize is won:
(1) The four-digit number is greater than 6400
(2) The four-digit number contains two digits 8 For example: If the four cards drawn are numbered $5, 8, 2, 8$ in order, then the four-digit number is 5828, and a prize is won. According to the above rules, there are (11-1)(11-2)(11-3)(11-4) four-digit numbers that can win prizes.

Consider all sequences composed of only the three digits 0, 1, 2. The length $n$ of a sequence refers to the sequence consisting of $n$ digits (which may repeat). Let $a(n)$ be the total count of consecutive pairs of zeros (i.e., 00) appearing in all sequences of length $n$. For example, among sequences of length 3 containing consecutive zeros, there are 000, 001, 002, 100, 200. Among these, 000 contributes 2 occurrences of 00, and each of the others contributes 1 occurrence of 00, so $a(3) = 6$. The value of $a(5)$ is $\square$.

On a circle, 12 equally spaced points are marked and numbered 1 to 12 in clockwise order. Among all triangles formed by choosing any 3 of these 12 points as vertices, the number of triangles whose three interior angles, arranged from smallest to largest, form an arithmetic sequence is (17-1)(17-2).

A school is holding a concert with 5 piano performances, 4 violin performances, and 3 vocal performances, totaling 12 different pieces. The school wants to arrange performances of the same type together, and vocal performances must come after either piano or violin performances. How many possible arrangements of pieces are there for this concert?
(1) $5 ! \times 4 ! \times 3 !$
(2) $2 \times 5 ! \times 4 ! \times 3 !$
(3) $3 \times 5 ! \times 4 ! \times 3 !$
(4) $4 \times 5 ! \times 4 ! \times 3 !$
(5) $6 \times 5 ! \times 4 ! \times 3 !$

A washing machine cycle must select one from 5 different fabric types (1, 2, 3, 4, 5), paired with one of 4 different modes (A, B, C, D), and there are 3 additional functions (A, B, C) that can be freely chosen to enable or disable. However, ``fabric type 1'' cannot be used simultaneously with additional function ``A''. For example, ``fabric type 2'' paired with ``mode A'', with both ``A'' and ``B'' additional functions enabled is a valid cycle; but ``fabric type 1'' paired with ``mode A'', with both ``A'' and ``B'' additional functions enabled is not a valid cycle. Based on the above, this washing machine has how many valid cycles?

A company hires 8 new employees, including 2 translators, 3 engineers, and 3 assistants. These 8 people are assigned to research and testing departments, with 4 people assigned to each department. Each department must include 1 translator and at least 1 engineer. There are (15--1)(15--2) ways to make such assignments.

There are $n$ children queuing in a line. You have $m$ candies and will begin handing out 1 or 2 candies to each child, starting from the first child in the line. You hand out the candies until reaching the end of the line or until there are no candies left. Answer the following questions. Note that $n$ and $m$ are positive integers.
I. Show the number of distribution patterns of candies if $n = m = 4$.
II. Show the number of distribution patterns of candies if $m \geq 2 n$.
III. Define $X _ { m }$ as the number of distribution patterns of candies if $n \geq m$. Show the recurrence formula satisfied by $X _ { m }$.
IV. Obtain $X _ { m }$ using the recurrence formula in Question III.
V. Consider the situation where the number of children is larger than the number of candies. Define $P ( m )$ as the ratio of the number of distribution patterns (where the distribution finishes by giving 2 candies) to the total number of distribution patterns. $P ( m )$ converges as $m$ increases. Compute the convergence value.
VI. Consider the situation where $m \geq 2 n$. The following rules are added to the way of handing out the candies: For the first child in the line, the probability of receiving 1 candy is $1 / 2$ and the probability of receiving 2 candies is $1 / 2$. If a child receives 1 candy, the probability of the next child receiving 1 candy is $1 / 2$ and the probability of receiving 2 candies is $1 / 2$. If a child receives 2 candies, the probability of the next child receiving 1 candy is $3 / 4$ and the probability of receiving 2 candies is $1 / 4$. Compute the probability that the $n$-th child in the line receives 2 candies.

For two-digit positive integers a and b
$$\frac { a ! } { b ! } = 132$$
Given this, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 22
B) 23
C) 24
D) 25
E) 26

$$( n + 2 ) ! - ( n + 1 ) ! - n ! = 2 ^ { 3 } \cdot 3 \cdot 5 ^ { 2 } \cdot 7$$
Given this, what is $n$?
A) 5
B) 6
C) 7
D) 8
E) 9

Three domestic automobiles of brands A, B, and C and three foreign automobiles of brands X, Y, and Z will be displayed in a single row at an exhibition according to the following conditions.

• Domestic and foreign automobiles will be arranged consecutively within their own groups.
• Brand A automobile will be in the first or last position among all automobiles.
• Brand X automobile will be in the first or last position among foreign automobiles.

Given this, in how many different ways can the automobiles be displayed?
A) 10
B) 12
C) 14
D) 16
E) 18

$$\frac { ( 10 ! ) ^ { 2 } - ( 9 ! ) ^ { 2 } } { 11 ! - 10 ! - 9 ! }$$
Which of the following is this operation equal to?
A) $8 !$
B) $9 !$
C) $10 !$
D) $8 \cdot 8 !$
E) $8 \cdot 9 !$

$$\frac { ( n + 1 ) ! + ( n - 1 ) ! } { n ^ { 3 } - 1 } = 24$$
Given this equality, what is n?
A) 3
B) 5
C) 6
D) 8
E) 9

$\frac { 6 ! + 7 ! } { ( 4 ! ) ^ { 2 } }$\ What is the result of this operation?\ A) 10\ B) 12\ C) 15\ D) 18\ E) 20

In a tournament with 8 teams, each team played against the other teams once. In the tournament, 3 referees were assigned from 4 available referees for each match, and all referees worked an equal number of matches.
Accordingly, how many matches did each referee work?
A) 14 B) 15 C) 18 D) 20 E) 21

Six friends named Ayça, Büşra, Ceyda, Deniz, Erdem and Furkan attending a party have a table with 6 chairs around it as shown in the figure.
Ayça and Büşra, who are on bad terms, do not want to sit in chairs that are next to each other or facing each other at this table. Accordingly, in how many different ways can these six friends sit in these chairs around the table?
A) 432
B) 384
C) 360
D) 288
E) 240

An airline has a total of 8 cabin crew members with different work experience for one morning and one evening flight to be performed.
Each of these employees will be on only one team, and two four-person flight teams will be formed from these employees such that the three most experienced employees are not on the same team.
Accordingly, in how many different ways can the morning and evening flight teams be formed?
A) 48
B) 54
C) 56
D) 60
E) 64