How many ordered pairs $( A , B )$ of disjoint subsets of the set $\{ 1,2,3,4,5,6 \}$ are there? [3 points]
(1) 729
(2) 720
(3) 243
(4) 64
(5) 36

As shown in the figure, a rectangular solid is made from 12 transparent glass boxes in the shape of identical cubes. If 4 of these glass boxes are replaced with glass boxes of the same size but black in color such that the rectangular solid viewed from above looks like (가) and viewed from the side looks like (나), how many ways can this be done? [4 points]
(1) 54
(2) 48
(3) 42
(4) 36
(5) 30

As shown in the figure, a rectangular solid is made from 12 transparent glass boxes in the shape of cubes of equal size. When 4 of these glass boxes are replaced with black glass boxes of the same size, and the view from above looks like (가) and the side view looks like (나), how many ways can this be done? [4 points]
(1) 54
(2) 48
(3) 42
(4) 36
(5) 30

Two adults and three children go to an amusement park to ride a certain ride. This ride has 2 chairs in the front row and 3 chairs in the back row. When each child must sit in the same row as an adult, find the number of ways for all 5 people to sit in the chairs of the ride. [4 points]

In how many ways can 9 identical candies be distributed into 5 identical bags such that no bag is empty? [3 points]
(1) 8
(2) 7
(3) 6
(4) 5
(5) 4

A music concert is divided into Part 1 and Part 2, with 2 solo teams, 2 ensemble teams, and 3 choir teams performing in total. The performance order of the 7 teams is to be determined according to the following two conditions.
(A) In Part 1, 3 teams perform in the order: solo, ensemble, choir.
(B) In Part 2, 4 teams perform in the order: solo, ensemble, choir, choir. What is the number of ways to determine the performance order for this music concert? [3 points]
(1) 18
(2) 20
(3) 22
(4) 24
(5) 26

A square is divided into three equal parts horizontally to create [Figure 1], and divided into three equal parts vertically to create [Figure 2]. [Figure 1] and [Figure 2] are alternately attached repeatedly to create the following figure. As shown in the figure, let A be the upper left vertex of the first attached [Figure 1], and let $\mathrm { B } _ { n }$ be the lower right vertex of the figure created by attaching a total of $n$ figures (combining the number of [Figure 1] and [Figure 2]).
When $a _ { n }$ is the number of shortest paths from vertex A to vertex $\mathrm { B } _ { n }$ along the lines, what is the value of $a _ { 3 } + a _ { 7 }$? [4 points]
(1) 26
(2) 28
(3) 30
(4) 32
(5) 34

A certain volunteer service center operates the following 4 volunteer activity programs every day.

Program	A	B	C	D
Volunteer Activity Hours	1 hour	2 hours	3 hours	4 hours

Chulsu wants to participate in one program each day for 5 days at this volunteer service center and create a volunteer activity plan so that the total volunteer activity hours is 8 hours. How many different volunteer activity plans can be created? [4 points]
(1) 47
(2) 44
(3) 41
(4) 38
(5) 35

There is a walking path in a rectangular lawn. As shown in the figure, this walking path consists of 8 circles with equal radii that are externally tangent to each other.
Starting from point A and arriving at point B along the walking path by the shortest distance, find the number of possible routes. (Note: The points marked on the circles represent the points of tangency between the circles and the rectangle or between the circles.) [4 points]

A company employee has 6 types of tasks to handle, including tasks $\mathrm { A }$ and $\mathrm { B }$. The employee wants to handle 4 types of tasks today, including $\mathrm { A }$ and $\mathrm { B }$, and task $\mathrm { A }$ must be handled before task $\mathrm { B }$. What is the number of ways to select the tasks to handle today and determine the order of handling the selected tasks? [3 points]
(1) 60
(2) 66
(3) 72
(4) 78
(5) 84

A company employee has 6 tasks to handle in total, including A and B. The employee wants to handle 4 tasks including A and B today, and A must be handled before B. What is the number of ways to select the tasks to handle today and determine the order of handling the selected tasks? [3 points]
(1) 60
(2) 66
(3) 72
(4) 78
(5) 84

There is a computer game where two dolls A and B are dressed in shirts and pants with undetermined colors, and then the colors of the clothes are determined. There are 3 shirts and 3 pants of different shapes, and the color of each piece of clothing is determined to be either red or green. A shirt and pants put on one doll are not put on the other doll. The colors of doll A's shirt and pants are determined to be different, and the colors of doll B's shirt and pants are also determined to be different. In this game, when dressing dolls A and B in shirts and pants and determining their colors, what is the number of possible outcomes? [4 points]
(1) 252
(2) 216
(3) 180
(4) 144
(5) 108

Find the natural number $n$ that satisfies the equation $2 \times {}_{n}\mathrm{C}_{3} = 3 \times {}_{n}\mathrm{P}_{2}$. [3 points]

Among the partitions of the natural number 7, how many distinct partitions can be expressed as the sum of natural numbers not exceeding 3? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For two natural numbers $m , n$ with $m, n \leq 20$, how many ordered pairs $( m , n )$ satisfy $f ( m n ) = f ( m ) + f ( n )$?
(1) 220
(2) 230
(3) 240
(4) 250
(5) 260

When a coin is tossed 5 times, what is the probability that the product of the number of heads and the number of tails is 6? [3 points]
(1) $\frac { 5 } { 8 }$
(2) $\frac { 9 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 3 } { 8 }$

Among the numbers $1,2,3,4,5$, if we select four numbers with repetition allowed and arrange them in a row to form a four-digit natural number that is a multiple of 5, how many cases are there? [3 points]
(1) 115
(2) 120
(3) 125
(4) 130
(5) 135

Find the value of ${ } _ { 5 } \mathrm { P } _ { 2 } + { } _ { 5 } \mathrm { C } _ { 2 }$. [3 points]

Find the total number of ordered triples $( a , b , c )$ of non-negative integers satisfying the following conditions. [4 points]
(a) $a + b + c = 7$
(b) $2 ^ { a } \times 4 ^ { b }$ is a multiple of 8.

When distributing 4 distinct balls into 4 distinct boxes without remainder, how many ways are there to distribute them such that there is at least one box containing exactly 1 ball? (Here, there may be boxes with no balls.) [4 points]
(1) 220
(2) 216
(3) 212
(4) 208
(5) 204

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

Find the value of ${ } _ { 6 } \mathrm { P } _ { 2 } - { } _ { 6 } \mathrm { C } _ { 2 }$. [3 points]

Find the value of ${}_{6}\mathrm{P}_{2} - {}_{6}\mathrm{C}_{2}$. [3 points]

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]

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