A training center operates 5 different types of experience programs. Two participants A and B, who participated in the programs at this training center, each want to select 2 types from the 5 types of experience programs. Find the number of cases where A and B select exactly one type of experience program in common. [4 points]

A training center operates five different types of experience programs. Participants A and B each want to select 2 types from the 5 types of experience programs. Find the number of cases where A and B select exactly one type in common. [4 points]

(Probability and Statistics) There are 9 balls in a bag, each labeled with a natural number from 1 to 9. When 4 balls are randomly drawn simultaneously from the bag, what is the probability that the sum of the largest and smallest numbers on the drawn balls is at least 7 and at most 9? [3 points]
(1) $\frac{5}{9}$
(2) $\frac{1}{2}$
(3) $\frac{4}{9}$
(4) $\frac{7}{18}$
(5) $\frac{1}{3}$

At a certain event venue, there are 5 locations where one banner can be installed at each location. There are three types of banners: A, B, and C, with 1 banner of type A, 4 banners of type B, and 2 banners of type C. When selecting and installing 5 banners at the 5 locations to satisfy the following conditions, how many possible outcomes are there? (Note: banners of the same type are not distinguished from each other.) [3 points]
(a) Banner A must be installed.
(b) Banner B is installed in at least 2 locations.
(1) 55
(2) 65
(3) 75
(4) 85
(5) 95

At a certain event venue, there are 5 locations where one banner can be installed at each location. There are three types of banners: A, B, and C, with 1 banner of type A, 4 banners of type B, and 2 banners of type C. When selecting and installing 5 banners at the 5 locations to satisfy the following conditions, how many possible cases are there? (Note: Banners of the same type are not distinguished from each other.) [3 points] (가) Banner A must be installed. (나) Banner B must be installed in at least 2 locations.
(1) 55
(2) 65
(3) 75
(4) 85
(5) 95

There are 2 students each from Korea, China, and Japan. When these 6 students each randomly select and sit in one of 6 seats with assigned seat numbers as shown in the figure, what is the probability that the two students from the same country sit such that the difference in their seat numbers is 1 or 10? [4 points]

11

12

13

21

22

23

(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 3 } { 20 }$
(4) $\frac { 1 } { 5 }$
(5) $\frac { 1 } { 4 }$

When 6 different balls are placed 3 each in two baskets A and B, how many possible outcomes are there? [3 points]

In a table tennis competition with 4 male table tennis players and 4 female table tennis players, when 2 people are randomly selected to form 4 teams, what is the probability that exactly 2 teams consist of 1 male and 1 female? [3 points]
(1) $\frac { 3 } { 7 }$
(2) $\frac { 18 } { 35 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 24 } { 35 }$
(5) $\frac { 27 } { 35 }$

When arranging 5 white flags and 5 blue flags in a line, how many ways are there to place white flags at both ends? (Note: flags of the same color are indistinguishable from each other.) [3 points]
(1) 56
(2) 63
(3) 70
(4) 77
(5) 84

For a natural number $r$, when ${}_{3}\mathrm{H}_{r} = {}_{7}\mathrm{C}_{2}$, find the value of ${}_{5}\mathrm{H}_{r}$. [3 points]

As shown in the figure, there is a road network connected in a diamond shape. Starting from point A and traveling the shortest distance to point B without passing through point C or point D, how many ways are there? [3 points]
(1) 26
(2) 24
(3) 22
(4) 20
(5) 18

In how many ways can 4 bottles of the same type of juice, 2 bottles of the same type of water, and 1 bottle of milk be distributed to 3 people without remainder? (Note: Some people may not receive any bottles.) [3 points]
(1) 330
(2) 315
(3) 300
(4) 285
(5) 270

In the following seating chart, 4 female students and 4 male students are randomly assigned to 8 seats excluding the seat at row 2, column 2, with one person per seat. Find the value of $70p$, where $p$ is the probability that at least 2 male students are seated adjacent to each other. (Two people are considered adjacent if they are next to each other in the same row or directly in front or behind each other in the same column.) [4 points]

When selecting 5 numbers from the digits $1,2,3,4$ with repetition allowed, how many cases are there where the digit 4 appears at most once? [3 points]
(1) 45
(2) 42
(3) 39
(4) 36
(5) 33

There are 8 white ping-pong balls and 7 orange ping-pong balls to be distributed entirely among 3 students. In how many ways can the balls be distributed so that each student receives at least one white ball and at least one orange ball? [4 points]
(1) 295
(2) 300
(3) 305
(4) 310
(5) 315

How many ordered pairs $( x , y , z , w )$ of non-negative integers satisfy the system of equations $$\left\{ \begin{array} { l } x + y + z + 3 w = 14 \\ x + y + z + w = 10 \end{array} \right.$$ ? [4 points]
(1) 40
(2) 45
(3) 50
(4) 55
(5) 60

Find the number of all ordered pairs $( a , b , c )$ of natural numbers satisfying the following conditions. [4 points] (가) $a \times b \times c$ is odd. (나) $a \leq b \leq c \leq 20$

For three integers $a , b , c$ satisfying $$1 \leq | a | \leq | b | \leq | c | \leq 5$$ what is the number of all ordered pairs $( a , b , c )$? [4 points]
(1) 360
(2) 320
(3) 280
(4) 240
(5) 200

How many ordered pairs $( a , b , c , d , e )$ of non-negative integers satisfy the following conditions? [4 points] (가) Among $a , b , c , d , e$, the number of 0's is 2. (나) $a + b + c + d + e = 10$
(1) 240
(2) 280
(3) 320
(4) 360
(5) 400

A jump is defined as moving from a point $( x , y )$ on the coordinate plane to one of the three points $( x + 1 , y )$, $( x , y + 1 ) , ( x + 1 , y + 1 )$. Let $X$ be the random variable representing the number of jumps when randomly selecting one case from all possible ways to move from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. The following is the process of finding the expected value $\mathrm { E } ( X )$ of the random variable $X$. (Here, each case is selected with equal probability.)
Let $N$ be the total number of ways to move from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. If the smallest value that the random variable $X$ can take is $k$, then $k =$ (a), and the largest value is $k + 3$.
$$\begin{aligned} & \mathrm { P } ( X = k ) = \frac { 1 } { N } \times \frac { 4 ! } { 3 ! } = \frac { 4 } { N } \\ & \mathrm { P } ( X = k + 1 ) = \frac { 1 } { N } \times \frac { 5 ! } { 2 ! 2 ! } = \frac { 30 } { N } \\ & \mathrm { P } ( X = k + 2 ) = \frac { 1 } { N } \times \text { (b) } \\ & \mathrm { P } ( X = k + 3 ) = \frac { 1 } { N } \times \frac { 7 ! } { 3 ! 4 ! } = \frac { 35 } { N } \end{aligned}$$
and $$\sum _ { i = k } ^ { k + 3 } \mathrm { P } ( X = i ) = 1$$ so $N =$ (c). Therefore, the expected value $\mathrm { E } ( X )$ of the random variable $X$ is as follows. $$\mathrm { E } ( X ) = \sum _ { i = k } ^ { k + 3 } \{ i \times \mathrm { P } ( X = i ) \} = \frac { 257 } { 43 }$$
When the numbers corresponding to (a), (b), (c) are $a , b , c$ respectively, what is the value of $a + b + c$? [4 points]
(1) 190
(2) 193
(3) 196
(4) 199
(5) 202

A jump is defined as moving from a point $( x , y )$ on the coordinate plane to one of the three points $( x + 1 , y )$, $( x , y + 1 )$, or $( x + 1 , y + 1 )$. Let $X$ be the random variable representing the number of jumps that occur when one case is randomly selected from all cases of moving from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. The following is the process of finding the mean $\mathrm { E } ( X )$ of the random variable $X$. (Here, each case is selected with equal probability.)
Let $N$ be the total number of cases of moving from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. If the smallest value that the random variable $X$ can take is $k$, then $k =$ (가), and the largest value is $k + 3$.
$$\begin{aligned} & \mathrm { P } ( X = k ) = \frac { 1 } { N } \times \frac { 4 ! } { 3 ! } = \frac { 4 } { N } \\ & \mathrm { P } ( X = k + 1 ) = \frac { 1 } { N } \times \frac { 5 ! } { 2 ! 2 ! } = \frac { 30 } { N } \\ & \mathrm { P } ( X = k + 2 ) = \frac { 1 } { N } \times \text { (나) } \\ & \mathrm { P } ( X = k + 3 ) = \frac { 1 } { N } \times \frac { 7 ! } { 3 ! 4 ! } = \frac { 35 } { N } \end{aligned}$$
and $$\sum _ { i = k } ^ { k + 3 } \mathrm { P } ( X = i ) = 1$$ so $N =$ (다). Therefore, the mean $\mathrm { E } ( X )$ of the random variable $X$ is as follows. $$\mathrm { E } ( X ) = \sum _ { i = k } ^ { k + 3 } \{ i \times \mathrm { P } ( X = i ) \} = \frac { 257 } { 43 }$$
When the numbers that fit (가), (나), and (다) are $a$, $b$, and $c$, respectively, what is the value of $a + b + c$? [4 points]
(1) 190
(2) 193
(3) 196
(4) 199
(5) 202

Find the value of ${}_{4}\mathrm{H}_{2}$. [3 points]

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

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

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