A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$.
Find the number of good paths from $(0,0)$ to $(6,6)$ that pass through both the points $(1,4)$ and $(2,3)$. [1 point]

A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$.
Find the number of good paths from $(0,0)$ to $(6,6)$ that pass through both the points $(1,2)$ and $(3,4)$. [2 points]

A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$.
Find the number of good paths from $(0,0)$ to $(6,6)$ such that neither of the two points $(1,2)$ and $(3,4)$ occurs on the path, i.e., the path must miss both of the points $(1,2)$ and $(3,4)$. [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

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

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

An up-right path is a sequence of points $\mathbf { a } _ { 0 } = \left( x _ { 0 } , y _ { 0 } \right) , \mathbf { a } _ { 1 } = \left( x _ { 1 } , y _ { 1 } \right) , \mathbf { a } _ { 2 } = ( x _ { 2 } , y _ { 2 } ), \ldots$ such that $\mathbf { a } _ { i + 1 } - \mathbf { a } _ { i }$ is either $( 1,0 )$ or $( 0,1 )$. The number of up-right paths from $( 0,0 )$ to $( 100,100 )$ which pass through $( 1,2 )$ is:
(A) $3 \cdot \binom { 197 } { 99 }$
(B) $3 \cdot \binom { 100 } { 50 }$
(C) $2 \cdot \binom { 197 } { 98 }$
(D) $3 \cdot \binom { 197 } { 100 }$.

15. A rectangle with sides ( $2 n - 1$ ) and ( $2 m - 1$ ) is divided into squares of unit length. The number of rectangle which can be formed with sides of odd length is :
(a) $m ^ { 2 } n ^ { 2 }$
(b) $m n ( m + 1 ) ( n + 1 )$
(c) $4 ^ { m + n - 1 }$
(d) none of these