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 $(4 ; 5 ; 1)$.

\begin{enumerate}
  \item Determine the number of possible draws.
  \item \begin{enumerate}
    \item[a.] Determine the number of draws without repetition of numbers.
    \item[b.] Deduce from this the number of draws containing at least one repetition of numbers.
  \end{enumerate}
\end{enumerate}

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.

\begin{enumerate}
  \setcounter{enumi}{2}
  \item Establish the probability distribution of the random variable $X_1$.
  \item Determine the expectation of the random variable $X_1$.
\end{enumerate}

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

\begin{enumerate}
  \setcounter{enumi}{4}
  \item Determine the expectation of the random variable $S$.
  \item Determine $P(S = 24)$.
  \item If a player obtains a sum greater than or equal to 22, then they win a prize.
  \begin{enumerate}
    \item[a.] Justify that there are exactly 10 draws allowing one to win a prize.
    \item[b.] Deduce from this the probability of winning a prize.
  \end{enumerate}
\end{enumerate}