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

A bag contains the following eight letters: A B C D E F G H (2 vowels and 6 consonants).
A game consists of drawing simultaneously at random two letters from this bag. You win if the draw consists of one vowel and one consonant.

1. A player draws simultaneously two letters from the bag. a. Determine the number of possible draws. b. Determine the probability that the player wins this game.

Questions 2 and 3 of this exercise are independent.

For the rest of the exercise, we admit that the probability that the player wins is equal to $\frac{3}{7}$.

1. To play, the player must pay $k$ euros, where $k$ is a non-zero natural integer. If the player wins, he receives 10 euros, otherwise he receives nothing. We denote $G$ the random variable equal to the algebraic gain of a player (that is, the sum received minus the sum paid). a. Determine the probability distribution of $G$. b. What must be the maximum value of the sum paid at the start for the game to remain favourable to the player?
2. Ten players each play one game. The letters drawn are returned to the bag after each game. We denote $X$ the random variable equal to the number of winning players. a. Justify that $X$ follows a binomial distribution and give its parameters. b. Calculate the probability, rounded to $10^{-3}$, that there are exactly four winning players. c. Calculate $P(X \geqslant 5)$ by rounding to $10^{-3}$. Give an interpretation of the result obtained. d. Determine the smallest natural integer $n$ such that $P(X \leqslant n) \geqslant 0.9$.

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. In a class of 24 students, there are 14 girls and 10 boys.
   Statement 1: It is possible to form 272 different groups of four students composed of two girls and two boys.
   
2. Let $f$ be the function defined on $\mathbb { R }$ by $f ( x ) = 3 \sin ( 2 x + \pi )$ and $C$ its representative curve in a given coordinate system.
   Statement 2: An equation of the tangent line to $C$ at the point with abscissa $\frac { \pi } { 2 }$ is $y = 6 x - 3 \pi$.
   
3. We consider the function $F$ defined on $] 0$; $+ \infty [$ by $F ( x ) = ( 2 x + 1 ) \ln ( x )$.
   Statement 3: The function $F$ is an antiderivative of the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = \frac { 2 } { x }$.
   
4. We consider the function $g$ defined on $\mathbb { R }$ by $g ( t ) = 45 \mathrm { e } ^ { 0.06 t } + 20$.
   Statement 4: The function $g$ is the unique solution of the differential equation $\left( E _ { 1 } \right) y ^ { \prime } + 0.06 y = 1.2$ satisfying $g ( 0 ) = 65$.
   
5. We consider the differential equation: $$\left( E _ { 2 } \right) : \quad y ^ { \prime } - y = 3 \mathrm { e } ^ { 0.4 x }$$ where $y$ is a positive function of the real variable $x$, defined and differentiable on $\mathbb { R }$ and $y ^ { \prime }$ the derivative function of the function $y$.
   Statement 5: The solutions of the equation $\left( E _ { 2 } \right)$ are convex functions on $\mathbb { R }$.

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

O valor de $\binom{10}{3}$ é
(A) 60 (B) 90 (C) 120 (D) 150 (E) 180

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 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

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

A father bought eight different gifts (among which, a bicycle and a cell phone) to give to his three children. He intends to distribute the gifts so that the oldest and youngest children receive three gifts each, and the middle one receives the two remaining gifts. The oldest will receive, among his gifts, either a bicycle or a cell phone, but not both.
In how many distinct ways can the distribution of gifts be made?
(A) 36
(B) 53
(C) 300
(D) 360
(E) 560

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.

How many non-congruent triangles are there with integer lengths $a \leq b \leq c$ such that $a + b + c = 20$?

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]

10. In how many ways can 10 identical chocolate bars be distributed among 5 children, in such a way that each child gets at least one chocolate bar?
(a) 50
(b) 126
(c) 252
(d) 3125

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

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

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

Five balls labeled with the numbers $1,2,3,4,5$ are to be placed into 3 boxes A, B, C. In how many ways can the balls be placed so that no box contains balls whose numbers sum to 13 or more? (Here, for an empty box, the sum of the numbers on the balls is taken as 0.) [4 points]
(1) 233
(2) 228
(3) 222
(4) 215
(5) 211