Question 5: An unknown code consists of 8 characters. Each character can be a letter or a digit. There are therefore 36 usable characters for each position.
A code-breaking software tests approximately one hundred million codes per second. In how much time at most can the software discover the code?

\begin{tabular}{l} a. approximately 0.3

seconds

& b. approximately 8 hours & c. approximately 3 hours &

d. approximately 470

hours

\hline \end{tabular}

An opaque bag contains eight tokens numbered from 1 to 8, indistinguishable to the touch. Three times, a player draws a token from this bag, notes its number, then puts it back in the bag. In this context, we call a ``draw'' the ordered list of the three numbers obtained. For example, if the player draws token number 4, then token number 5, then token number 1, then the corresponding draw is $(4 ; 5 ; 1)$.

1. Determine the number of possible draws.
2. 1. [a.] Determine the number of draws without repetition of numbers.
   2. [b.] Deduce from this the number of draws containing at least one repetition of numbers.

We denote $X_1$ the random variable equal to the number of the first token drawn, $X_2$ the one equal to the number of the second token drawn and $X_3$ the one equal to the number of the third token drawn. Since this is a draw with replacement, the random variables $X_1, X_2$, and $X_3$ are independent and follow the same probability distribution.

1. Establish the probability distribution of the random variable $X_1$.
2. Determine the expectation of the random variable $X_1$.

We denote $S = X_1 + X_2 + X_3$ the random variable equal to the sum of the numbers of the three tokens drawn.

1. Determine the expectation of the random variable $S$.
2. Determine $P(S = 24)$.
3. If a player obtains a sum greater than or equal to 22, then they win a prize.
   1. [a.] Justify that there are exactly 10 draws allowing one to win a prize.
   2. [b.] Deduce from this the probability of winning a prize.

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.

Exercise 3

The ``base64'' encoding, used in computing, allows messages and other data such as images to be represented and transmitted using 64 characters: the 26 uppercase letters, the 26 lowercase letters, the digits 0 to 9 and two other special characters. Parts A, B and C are independent.

Part A

In this part, we are interested in sequences of 4 characters in base64. For example, ``gP3g'' is such a sequence. In a sequence, order must be taken into account: the sequences ``m5C2'' and ``5C2m'' are not identical.

1. Determine the number of possible sequences.
2. Determine the number of sequences if we require that the 4 characters are pairwise different.
3. a. Determine the number of sequences containing no uppercase letter A. b. Deduce the number of sequences containing at least one uppercase letter A. c. Determine the number of sequences containing exactly one uppercase letter A. d. Determine the number of sequences containing exactly two uppercase letters A.

Part B

We are interested in the transmission of a sequence of 250 characters from one computer to another. We assume that the probability that a character is incorrectly transmitted is equal to 0.01 and that the transmissions of the different characters are independent of each other. We denote by $X$ the random variable equal to the number of characters incorrectly transmitted.

1. We admit that the random variable $X$ follows the binomial distribution. Give its parameters.
2. Determine the probability that all characters are correctly transmitted. The exact expression will be given, then an approximate value to $10^{-3}$ near.
3. What do you think of the following statement: ``The probability that more than 16 characters are incorrectly transmitted is negligible''?

Part C

We are now interested in the transmission of 4 sequences of 250 characters. We denote by $X_1, X_2, X_3$ and $X_4$ the random variables corresponding to the numbers of characters incorrectly transmitted during the transmission of each of the 4 sequences. We admit that the random variables $X_1, X_2, X_3$ and $X_4$ are independent of each other and follow the same distribution as the random variable $X$ defined in part B. We denote by $S = X_1 + X_2 + X_3 + X_4$. Determine, by justifying, the expectation and the variance of the random variable $S$.

Exercise 4
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
A museum offers visits with or without an audioguide. Tickets can be purchased online or directly at the counter.

1. When a person buys their ticket online, a validation code is sent to them by SMS so they can confirm their purchase. This code is generated randomly and consists of 4 digits that are pairwise distinct, with the first digit being different from 0.
   Statement 1: The number of different codes that can be generated is 5040.
   
2. A study made it possible to consider that:
   • the probability that a person chooses the audioguide given that they bought their ticket online is equal to 0{,}8;
   • the probability that a person buys their ticket online is equal to 0{,}7;
   • the probability that a person opts for a visit without an audioguide is equal to 0{,}32.
   
   Statement 2: The probability that a visitor does not take the audioguide given that they bought their ticket at the counter is greater than two thirds.
   
3. We randomly choose 12 visitors to this museum.
   We assume that the choice of the ``audioguide'' option is independent from one visitor to another.
   Statement 3: The probability that exactly half of these visitors opt for the audioguide is equal to $924 \times 0{,}2176^6$.
   
4. When a person has an audioguide, they can choose from three routes:
   • a first one lasting fifty minutes,
   • a second one lasting one hour and twenty minutes,
   • a third one lasting one hour and forty minutes.
   
   The tour time can be modelled by a random variable $X$ whose probability distribution is given below:
   

   $x_i$	$50\,\min$	$1\,\mathrm{h}\,20\,\min$	$1\,\mathrm{h}\,40\,\min$
   $P(X = x_i)$	0{,}1	0{,}6	0{,}3

   
   Statement 4: The expectation of $X$ is 77 minutes.

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. Let $E$ and $F$ be the sets $E = \{1; 2; 3; 4; 5; 6; 7\}$ and $F = \{0; 1; 2; 3; 4; 5; 6; 7; 8; 9\}$. Statement $\mathbf{n^\circ 1}$: There are more 3-tuples of distinct elements of $E$ than 4-element combinations of $F$.
2. In the orthonormal coordinate system, we have represented the square function, denoted $f$, as well as the square ABCD with side 3. Statement $\mathbf{n^\circ 2}$: The shaded region and the square ABCD have the same area.
3. We consider the integral $J$ below: $$J = \int_1^2 x\ln(x)\,\mathrm{d}x$$ Statement $\mathbf{n^\circ 3}$: Integration by parts makes it possible to obtain: $J = \dfrac{7}{11}$.
4. On $\mathbb{R}$, we consider the differential equation $$(E): \quad y' = 2y - \mathrm{e}^x.$$ Statement $\mathbf{n^\circ 4}$: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x + \mathrm{e}^{2x}$ is a solution of the differential equation $(E)$.
5. Let $x$ be given in $[0; 1[$. We consider the sequence $(u_n)$ defined for any natural integer $n$ by: $$u_n = (x-1)\mathrm{e}^n + \cos(n).$$ Statement $\mathbf{n^\circ 5}$: The sequence $(u_n)$ diverges to $-\infty$.

O número de anagramas da palavra AMOR é
(A) 12 (B) 16 (C) 20 (D) 24 (E) 32

A bank asked its customers to create a personal six-digit password, formed only by digits from 0 to 9, for access to their checking account via the internet.
However, an expert in electronic security systems recommended to the bank's management to re-register its users, requesting, for each one of them, the creation of a new six-digit password, now allowing the use of the 26 letters of the alphabet, in addition to digits from 0 to 9. In this new system, each uppercase letter was considered distinct from its lowercase version. Furthermore, the use of other types of characters was prohibited.
One way to evaluate a change in the password system is to verify the improvement coefficient, which is the ratio of the new number of password possibilities to the old one.
The improvement coefficient of the recommended change is
(A) $\frac{62^{6}}{10^{6}}$ (B) $\frac{62!}{10!}$ (C) $\frac{62! \cdot 4!}{10! \cdot 56!}$ (D) $62! - 10!$ (E) $62^{6} - 10^{6}$

A jewellery artisan has at his disposal Brazilian stones of three colours: red, blue and green.
He intends to produce jewellery made of a metal alloy, based on a mould in the shape of a non-square rhombus with stones at its vertices, so that two consecutive vertices always have stones of different colours.
The figure illustrates a piece of jewellery, produced by this artisan, whose vertices $A$, $B$, $C$ and $D$ correspond to the positions occupied by the stones.
Based on the information provided, how many different pieces of jewellery, in this format, can the artisan obtain?
(A) 6 (B) 12 (C) 18 (D) 24 (E) 36

QUESTION 157
The number of ways to arrange 4 people in a row is
(A) 12
(B) 16
(C) 20
(D) 24
(E) 28

QUESTION 178
The value of $5! - 3!$ is
(A) 108
(B) 112
(C) 114
(D) 116
(E) 120

To stimulate his daughter's reasoning, a father made the following drawing and gave it to the child along with three colored pencils. He wants the girl to paint only the circles, so that those connected by a segment have different colors.
In how many different ways can the child do what the father asked?
(A) 6
(B) 12
(C) 18
(D) 24
(E) 72

A company will build its page on the internet and expects to attract an audience of approximately one million customers. To access this page, a password with a format to be defined by the company will be required. There are five format options offered by the programmer, described in the table, where ``L'' and ``D'' represent, respectively, uppercase letter and digit.

Option	Format
I	LDDDDD
II	DDDDDD
III	LLDDDD
IV	DDDDD
V	LLLDD

The letters of the alphabet, among the 26 possible ones, as well as the digits, among the 10 possible ones, can be repeated in any of the options.
The company wants to choose a format option whose number of possible distinct passwords is greater than the expected number of customers, but such that this number is not greater than twice the expected number of customers.
The option that best suits the company's conditions is
(A) I.
(B) II.
(C) III.
(D) IV.
(E) V.

How many ways can 4 people be arranged in a line?
(A) 8
(B) 12
(C) 16
(D) 20
(E) 24

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

Ten couples founded a dance group and decided to establish a board of directors with three positions: president, secretary, and treasurer. For greater representation, it was decided that at most one person per couple could hold a position on this board.
How many different boards can be formed by these 10 couples?
(A) $10 \times 9 \times 8$
(B) $20 \times 18 \times 16$
(C) $20 \times 19 \times 18$
(D) $10 \times 9 \times 8 \times 2$
(E) $20 \times 18 \times 16 \times 2$

The word MATHEMATICS consists of 11 letters. The number of distinct ways to rearrange these letters is
(A) $11 !$
(B) $\frac { 11 ! } { 3 }$
(C) $\frac { 11 ! } { 6 }$
(D) $\frac { 11 ! } { 8 }$

In a business meeting, each person shakes hands with each other person, with the exception of Mr. L. Since Mr. L arrives after some people have left, he shakes hands only with those present. If the total number of handshakes is exactly 100 , how many people left the meeting before Mr. L arrived? (Nobody shakes hands with the same person more than once.)

Let S be the set of all 5-digit numbers that contain the digits $1,3,5,7$ and 9 exactly once (in usual base 10 representation). Show that the sum of all elements of S is divisible by 11111. Find this sum.

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:

The format for car license plates in a small country is two digits followed by three vowels, e.g. 04 IOU. A license plate is called ``confusing'' if the digit 0 (zero) and the vowel O are both present on it. For example $04\,IOU$ is confusing but $20\,AEI$ is not. (i) How many distinct number plates are possible in all? (ii) How many of these are not confusing?

10 mangoes are to be placed in 5 distinct boxes labeled $\mathrm{U}, \mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y}$. A box may contain any number of mangoes including no mangoes or all the mangoes. What is the number of ways to distribute the mangoes so that exactly two of the boxes contain exactly two mangoes each?

A broken calculator has all its 10 digit keys and two operation keys intact. Let us call these operation keys A and B. When the calculator displays a number $n$ pressing A changes the display to $n+1$. When the calculator displays a number $n$ pressing $B$ changes the display to $2n$. For example, if the number 3 is displayed then the key strokes ABBA changes the display in the following steps $3 \rightarrow 4 \rightarrow 8 \rightarrow 16 \rightarrow 17$.
If 1 is on the display what is the least number of key strokes needed to get 260 on the display?

Let $\pi = \pi_{1}\pi_{2}\ldots\ldots\pi_{n}$ be a permutation of the numbers $1,2,3,\ldots,n$. We say $\pi$ has its first ascent at position $k < n$ if $\pi_{1} > \pi_{2} \ldots > \pi_{k}$ and $\pi_{k} < \pi_{k+1}$. If $\pi_{1} > \pi_{2} > \ldots > \pi_{n-1} > \pi_{n}$ we say $\pi$ has its first ascent in position $n$. For example when $n = 4$ the permutation 2134 has its first ascent at position 2.
The number of permutations which have their first ascent at position $k$ is ......

You are supposed to create a 7-character long password for your mobile device.
(i) How many 7-character passwords can be formed from the 10 digits and 26 letters? (Only lowercase letters are taken throughout the problem.) Repeats are allowed, e.g., 0001a1a is a valid password.
(ii) How many of the passwords contain at least one of the 26 letters and at least one of the 10 digits? Write your answer in the form: (Answer to part i) $-$ (something).
(iii) How many of the passwords contain at least one of the 5 vowels, at least one of the 21 consonants and at least one of the 10 digits? Extend your method for part ii to write a formula and explain your reasoning.
(iv) Now suppose that in addition to the lowercase letters and digits, you can also use 12 special characters. How many 7-character passwords are there that contain at least one of the 5 vowels, at least one of the 21 consonants, at least one of the 10 digits and at least one of the 12 special characters? Write only the final formula analogous to your answer to part iii. Do NOT explain.