A first-year general education student chooses three specializations from the twelve offered. The number of possible combinations is: a. 1728 b. 1320 c. 220 d. 33

An online gaming company offers a new smartphone application called ``Heart Tickets!''. Each participant generates on their smartphone a ticket containing a $3 \times 3$ grid on which three hearts are placed randomly. The ticket is winning if the three hearts are positioned side by side on the same line, on the same column or on the same diagonal.

1. Justify that there are exactly 84 different ways to position the three hearts on a grid.
2. Show that the probability that a ticket is winning equals $\frac{2}{21}$.
3. When a player generates a ticket, the company deducts \euro{}1 from their bank account. If the ticket is winning, the company then gives the player \euro{}5. Is the game favorable to the player?
4. A player decides to generate 20 tickets on this application. We assume that the generations of tickets are independent of each other. a. Give the probability distribution of the random variable $X$ which counts the number of winning tickets among the 20 tickets generated. b. Calculate the probability, rounded to $10^{-3}$, of the event $(X = 5)$. c. Calculate the probability, rounded to $10^{-3}$, of the event $(X \geqslant 1)$ and interpret the result in the context of the exercise.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. Two football teams of 22 and 25 players shake hands at the end of a match. Each player from one team shakes hands once with each player from the other team.

Statement 1 47 handshakes were exchanged.
2. A race involves 18 competitors. The three first-place finishers are rewarded indiscriminately by offering the same prize to each.
Statement 2 There are 4896 possibilities for distributing these prizes.
3. An association organizes a hurdle race competition that will establish a podium (the podium consists of the three best athletes ranked in their order of arrival). Seven athletes participate in the tournament. Jacques is one of them.
Statement 3 There are 90 different podiums on which Jacques appears.
4. Let $X _ { 1 }$ and $X _ { 2 }$ be two random variables with the same distribution given by the table below:

$x _ { i }$	- 2	- 1	2	5
$P \left( X = x _ { i } \right)$	0.1	0.4	0.3	0.2

We assume that $X _ { 1 }$ and $X _ { 2 }$ are independent and we consider $Y$ the random variable sum of these two random variables. Statement 4 $P ( Y = 4 ) = 0.25$.
5. A swimmer trains with the objective of swimming 50 metres freestyle in less than 25 seconds. Through training, it turns out that the probability of achieving this is 0.85. He performs, on one day, 20 timed 50-metre swims. We denote by $X$ the random variable that counts the number of times he swims this distance in less than 25 seconds on this day. We admit that $X$ follows the binomial distribution with parameters $n = 20$ and $p = 0.85$.
Statement 5 Given that he achieved his objective at least 15 times, an approximate value to $10 ^ { - 3 }$ of the probability that he achieved it at least 18 times is 0.434.

O número de diagonais de um polígono convexo de $n$ lados é dado pela fórmula $D = \dfrac{n(n-3)}{2}$. Um polígono convexo tem 20 diagonais. O número de lados desse polígono é
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

O número de combinações de 8 elementos tomados 3 a 3 é
(A) 24 (B) 40 (C) 56 (D) 112 (E) 336

Consider the following betting game:
On a ticket with 60 available numbers, a bettor chooses from 6 to 10 numbers. Among the available numbers, only 6 will be drawn. The bettor will be awarded if the 6 drawn numbers are among the numbers chosen by him on the same ticket.
The table presents the price of each ticket, according to the quantity of numbers chosen.

\begin{tabular}{ c } Quantity of numbers

chosen on a ticket

& Ticket price (R\$) \hline 6 & 2.00 \hline 7 & 12.00 \hline 8 & 40.00 \hline 9 & 125.00 \hline 10 & 250.00 \hline \end{tabular}
Five bettors, each with R\$ 500.00 to bet, made the following choices:
Arthur: 250 tickets with 6 numbers chosen; Bruno: 41 tickets with 7 numbers chosen and 4 tickets with 6 numbers chosen; Caio: 12 tickets with 8 numbers chosen and 10 tickets with 6 numbers chosen; Douglas: 4 tickets with 9 numbers chosen; Eduardo: 2 tickets with 10 numbers chosen.
The two bettors with the highest probabilities of being awarded are
(A) Caio and Eduardo. (B) Arthur and Eduardo. (C) Bruno and Caio. (D) Arthur and Bruno. (E) Douglas and Eduardo.

QUESTION 152
The number of diagonals of a polygon with 8 sides is
(A) 16
(B) 18
(C) 20
(D) 22
(E) 24

QUESTION 165
The number of combinations of 6 elements taken 2 at a time is
(A) 12
(B) 15
(C) 18
(D) 21
(E) 24

QUESTION 172
The value of $\binom{5}{2}$ is
(A) 5
(B) 8
(C) 10
(D) 12
(E) 15

A children's toy truck-carrier is formed by a trailer and ten small cars transported on it. In the production sector of the company that manufactures this toy, all the small cars are painted so that the toy looks more attractive. The colors used are yellow, white, orange and green, and each small car is painted with only one color. The truck-carrier has a fixed color. The company determined that in every truck-carrier there must be at least one small car of each of the four available colors. Change of position of the small cars on the truck-carrier does not generate a new model of the toy.
Based on this information, how many distinct models of the truck-carrier toy can this company produce?
(A) $C_{6,4}$
(B) $C_{9,3}$
(C) $C_{10,4}$
(D) $6^{4}$
(E) $4^{6}$

Not being fans of practicing sports, a group of friends decided to hold a soccer tournament using a video game. They decided that each player plays only once against each of the other players. The champion will be the one who gets the highest number of points. They observed that the number of matches played depends on the number of players, as shown in the table:

\begin{tabular}{ c } Number of

players

& 2 & 3 & 4 & 5 & 6 & 7 \hline

Number of

matches

& 1 & 3 & 6 & 10 & 15 & 21 \hline \end{tabular}
If the number of players is 8, how many matches will be played?
(A) 64
(B) 56
(C) 49
(D) 36
(E) 28

The number of diagonals of a polygon with 8 sides is:
(A) 16
(B) 18
(C) 20
(D) 22
(E) 24

A committee of 3 people is to be chosen from a group of 7. How many different committees are possible?
(A) 21
(B) 28
(C) 35
(D) 42
(E) 56

The value of $\binom{6}{2}$ is:
(A) 10
(B) 12
(C) 15
(D) 18
(E) 20

There are 8 boys and 7 girls in a group. For each of the tasks specified below, write an expression for the number of ways of doing it. Do NOT try to simplify your answers. a) Sitting in a row so that all boys sit contiguously and all girls sit contiguously, i.e., no girl sits between any two boys and no boy sits between any two girls
Answer: b) Sitting in a row so that between any two boys there is a girl and between any two girls there is a boy
Answer: c) Choosing a team of six people from the group
Answer: d) Choosing a team of six people consisting of unequal number of boys and girls
Answer:

A regular 100-sided polygon is inscribed in a circle. Suppose three of the 100 vertices are chosen at random, all such combinations being equally likely. Find the probability that the three chosen points form vertices of a right angled triangle.

[12 points] Let $N = \{ 1,2,3,4,5,6,7,8,9 \}$ and $L = \{ a , b , c \}$.
(i) Suppose we arrange the 12 elements of $L \cup N$ in a line such that no two of the three letters occur consecutively. If the order of the letters among themselves does not matter, find the number such arrangements.
(ii) Find the number of functions from $N$ to $L$ such that exactly 3 numbers are mapped to each of $a , b$ and $c$.
(iii) Find the number of onto functions from $N$ to $L$.

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]

Among 12-character strings made using all eight $a$'s and four $b$'s, how many strings satisfy all of the following conditions? [4 points]
(a) $b$ cannot appear consecutively.
(b) If the first character is $b$, then the last character is $a$.
(1) 70
(2) 105
(3) 140
(4) 175
(5) 210

Among 12-character strings made using all eight $a$'s and four $b$'s, how many strings satisfy all of the following conditions? [4 points] (가) $b$ cannot appear consecutively. (나) If the first character is $b$, then the last character is $a$.
(1) 70
(2) 105
(3) 140
(4) 175
(5) 210

When arranging $1, 2, 2, 4, 5, 5$ in a line to form a six-digit natural number, find the number of natural numbers greater than 300000. [4 points]

When selecting two different odd numbers from the odd numbers from 1 to 30, how many cases are there where the sum of the two numbers is a multiple of 3? [4 points]
(1) 43
(2) 41
(3) 39
(4) 37
(5) 35

Five balls labeled with the numbers $1,2,3,4,5$ are to be placed into 3 boxes $\mathrm { A } , \mathrm { B } , \mathrm { C }$. How many ways are there to place the balls in the boxes such that no box has a sum of the numbers on the balls that is 13 or more? (Note: For an empty box, the sum of the numbers on the balls is considered to be 0.) [4 points]
(1) 233
(2) 228
(3) 222
(4) 215
(5) 211