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.

Given set $A = \left\{ ( x , y ) \left| x ^ { 2 } + y ^ { 2 } \leqslant 3 , x \in \mathbf { Z } , y \in \mathbf { Z } \right. \right\}$, the number of elements in $A$ is
A. 9
B. 8
C. 5
D. 4

We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, an outcome resulting from $n$ successive draws is modeled by the $n$-tuple indicating the color and number of the balls successively obtained. We denote by $\Omega_{n}$ the set of possible outcomes of these $n$ draws.
By examining the number of balls in the urn just before each draw, justify that, for $n \geqslant 1$, $$\operatorname{card}(\Omega_{n}) = (a_{0} + b_{0}) \times \cdots \times (a_{0} + b_{0} + s(n-1)) = s^{n} L_{n}\left(\frac{a_{0} + b_{0}}{s}\right).$$

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players A, B, C, D, E and F play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, B has 8 points and C has 4 points. It is also known that E won against F. In the next set of games D, E and F win their games against A, B and C respectively. If A, B and D move to the final round, the final scores of E and F are, respectively,
(A) 4 and 2
(B) 2 and 4
(C) 2 and 2
(D) 4 and 4

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players A, B, C, D, E and F play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, B has 8 points and C has 4 points. It is also known that E won against F. In the next set of games D, E and F win their games against A, B and C respectively. If A, B and D move to the final round, the final scores of E and F are, respectively,
(A) 4 and 2
(B) 2 and 4
(C) 2 and 2
(D) 4 and 4

A party is attended by twenty people. In any subset of four people, there is at least one person who knows the other three (we assume that if $X$ knows $Y$, then $Y$ knows $X$). Suppose there are three people in the party who do not know each other. How many people in the party know everyone?
(A) 16
(B) 17
(C) 18
(D) Cannot be determined from the given data.

Let $|X|$ denote the number of elements in a set $X$. Let $S = \{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \leq |B| < |A|$, equals\_\_\_\_

Let $S = \left\{ E _ { 1 } , E _ { 2 } \ldots E _ { 8 } \right\}$ be a sample space of a random experiment such that $P \left( E _ { n } \right) = \frac { n } { 36 }$ for every $n = 1,2 \ldots 8$. Then the number of elements in the set $\left\{ A \subset S : P ( A ) \geq \frac { 4 } { 5 } \right\}$ is $\_\_\_\_$.

The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is $\underline{\hspace{1cm}}$.

Let $A = \{ 1,2,3,4,5 \}$. Let R be a relation on A defined by $x \mathrm { R } y$ if and only if $4 x \leq 5 \mathrm { y }$. Let m be the number of elements in R and n be the minimum number of elements from $\mathrm { A } \times \mathrm { A }$ that are required to be added to R to make it a symmetric relation. Then $\mathrm { m } + \mathrm { n }$ is equal to:
(1) 25
(2) 24
(3) 26
(4) 23

Let $\mathbf { R } = \{ ( 1,2 ) , ( 2,3 ) , ( 3,3 ) \}$ be a relation defined on the set $\{ 1,2,3,4 \}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:
(1) 10
(2) 7
(3) 8
(4) 9

$$\begin{aligned} & A = \{ a , b , e \} \\ & B = \{ a , b , c , d \} \end{aligned}$$
Given this, how many sets $K$ satisfy the condition $( A \cap B ) \subseteq K \subseteq ( A \cup B )$?
A) 3
B) 4
C) 5
D) 8
E) 9

Let $\mathbf { A } = \{ \mathbf { a } , \mathbf { b } , \mathbf { c } , \mathbf { d } \}$. For non-empty subsets $X , Y$ of A
$$\begin{aligned} & X \cap Y = \emptyset \\ & X \cup Y = A \end{aligned}$$
How many ordered pairs (X, Y) are there such that these conditions hold?
A) 6
B) 8
C) 10
D) 12
E) 14

The figure below shows a lamp and the appearance of a string that operates this lamp. The lamp;

• when closed, if the string is pulled and released, it gives dim light,
• when giving dim light, if the string is pulled and released, it gives daylight,
• when giving daylight, if the string is pulled and released, it gives bright light,
• when giving bright light, if the string is pulled and released, it turns off.

Initially, this lamp was closed. The string was pulled and released A times and the lamp was observed to give bright light. Then, the lamp's string was pulled and released B more times and the lamp was observed to give daylight. Later, the lamp's string was pulled and released C more times and the lamp was observed to turn off.
Accordingly, which of the following is an even number?
A) $A \cdot B + C$ B) $B \cdot C + A$ C) $A \cdot (B + C)$ D) $B \cdot (A + C)$ E) $C \cdot (A + B)$

At the entrance of a hotel, there are three digital wall clocks showing the local times of cities $\mathrm{A}, \mathrm{B}$ and C. A customer looking at these clocks observed that the local time difference between cities A and B is 4 hours, and the local time difference between cities B and C is 3 hours.
When the clock showing the local time of city A reads 14.00, which of the following cannot be the time shown on the clock for city C?
A) 07.00 B) 13.00 C) 15.00 D) 17.00 E) 21.00

The number of different digits in a natural number N is defined as shown in the figure.
Example: $4202 = 3$
Let A be a digit, and the equality shown in the figure holds.
What is the sum of different A values that satisfy the equality?
A) 8 B) 9 C) 10 D) 11 E) 12

Among the three-digit natural numbers $ABB$ and $BAB$, one is divisible by 11 and the other is divisible by 12.
Accordingly, what is the sum $\mathrm{A} + \mathrm{B}$?
A) 7 B) 8 C) 10 D) 11 E) 13

Let $A, B$ and $C$ be digits different from zero and from each other; the sum of the two-digit natural number AB and the two-digit natural number BC equals one less than the two-digit natural number CA.
Accordingly, how many different three-digit natural numbers ABC can be written using the digits A, B, and C that satisfy this condition?
A) 1 B) 2 C) 3 D) 5 E) 6

Boat trips are organized for visitors who want to visit a tourist island. The boat departs when there are at least 20 passengers and can carry a maximum of 35 passengers. On a particular day, 3 boat trips were organized and a total of 91 passengers were transported. The ratio of the number of passengers transported in the first trip to the number of passengers transported in the second trip is $\frac{4}{5}$.
Accordingly, how many passengers were transported in the third trip?
A) 21 B) 24 C) 28 D) 32 E) 35

Yeşim organized a poll on social media with a mosque photo she took in Edirne. After a certain period, the number of votes used for some cities and the ratio of the number of votes used for some cities to the total number of votes as a percentage are given. After this distribution, 5 more votes were cast in total, 4 of which were for Edirne and 1 for Istanbul.
Accordingly, what is the percentage of the number of votes cast for Edirne to the total number of votes in the final situation?
A) 36 B) 38 C) 40 D) 42 E) 44

The campaigns at stationery stores $A$ and $B$, which have the same prices for the same type of products, are as follows.

• At stationery store A, when a backpack is purchased, a pen case is sold at half price.
• At stationery store B, when 2 of the same type of product are purchased, a 40\% discount is applied to the 2nd product.

Ege and Deniz took advantage of stationery store A's campaign and each bought one backpack priced at 300 TL and one pen case priced at 120 TL.
If Ege and Deniz had each bought one backpack and one pen case of the same type from stationery store B, how much less would the total amount they paid be compared to what they paid to stationery store A?
A) 30 B) 36 C) 42 D) 48 E) 54

A painter displayed all of his paintings at his first exhibition and sold some of them. In all subsequent exhibitions, this painter displayed the paintings that were not sold at the previous exhibition together with new paintings he created.
The painter sold $\frac{3}{5}$ of the paintings he displayed at each exhibition. Also, for each exhibition after the first, he created as many new paintings as the number of paintings remaining from the previous exhibition.
Given that the painter sold 96 paintings at his 3rd exhibition, how many paintings did he display at his first exhibition?
A) 100 B) 150 C) 200 D) 250 E) 300

Aras divides all of his marbles into groups of 3 and obtains a two-digit natural number AB, and divides them into groups of 8 and obtains a two-digit natural number BA.
According to this, if Aras divides all the marbles he has into groups such that each group has an equal number of marbles, which of the following could be the number of groups he obtains?
A) 40 B) 48 C) 54 D) 56 E) 60

Serkan's wardrobe contains three types of clothing: shirt ($G$), pants ($P$), and jacket (C). The numerical distribution of these clothes initially in the wardrobe is shown in the pie chart below.
Serkan takes a certain number of clothes from his wardrobe to dry cleaning. In the final situation, the numerical distribution of the clothes remaining in Serkan's wardrobe shown in the pie chart is the same as the initial one.
Given that Serkan initially had 5 jackets in his wardrobe and he took 1 of these jackets to dry cleaning, how many shirts are left in Serkan's wardrobe?
A) 8 B) 10 C) 15 D) 18 E) 20

When Ali dede, whose grandfather's restaurant was founded in 1949, came to the restaurant, he said the following sentence to his grandson.
``In the year this restaurant was founded, I was 11 years old, and in the year you were born, this place had been in service for 40 years.''
Accordingly, what is the sum of the ages of Ali dede and his grandson in 2022?
A) 101 B) 105 C) 109 D) 113 E) 117