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.

0-1 periodic sequences have important applications in communication technology. If a sequence $a _ { 1 } a _ { 2 } \cdots a _ { n } \cdots$ satisfies $a _ { i } \in \{ 0,1 \} ( i = 1,2 , \cdots )$ and there exists a positive integer $m$ such that $a _ { i + m } = a _ { i } ( i = 1,2 , \cdots )$ , then it is called a 0-1 periodic sequence, and the smallest positive integer $m$ satisfying $a _ { i + m } = a _ { i } ( i = 1,2 , \cdots )$ is called the period of this sequence. For a 0-1 sequence $a _ { 1 } a _ { 2 } \cdots a _ { n } \cdots$ with period $m$ , $C ( k ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } a _ { i } a _ { i + k } ( k = 1,2 , \cdots , m - 1 )$ is an important index describing its properties. Among the following 0-1 sequences with period 5, the one satisfying $C ( k ) \leqslant \frac { 1 } { 5 } ( k = 1,2,3,4 )$ is
A． $11010 \ldots$
B． $11011 \cdots$
C． $10001 \cdots$
D． $11001 \cdots$

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$. We denote by $W_{n} = U_{n} \wedge V_{n}$ (the GCD of $U_n$ and $V_n$).
For all $k \in \mathbb{N}^{*}$, show that $$\mathbb{P}\left(W_{n} \in k\mathbb{N}^{*}\right) = \left(\frac{\lfloor n/k \rfloor}{n}\right)^{2}$$