\section*{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.

\section*{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.

\begin{enumerate}
  \item Determine the number of possible sequences.
  \item Determine the number of sequences if we require that the 4 characters are pairwise different.
  \item 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.
\end{enumerate}

\section*{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.

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

\section*{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 $S$.