Problem $2.$ A computer system has $n$ users, each with a unique name and password. Due to a

software error, the $n$ passwords are randomly permuted internally $($i$.$e$.$ each of the $n!$ possible

permutations are equally likely$).$ Only those users lucky enough to have had their passwords

unchanged in the permutation are able to continue using the system.

f$)$ a$)$ What is the probability that a particular user, say user $1,$ is able to continue using the

system?

g$)$ b$)$ What is the probability that $j$ users out of $n$ are still able to continue using the system?

Hint: Let $x_i$ be a $r.v.$ with the value $1$ if user i can use the system and $0$ otherwise.

h$)$ b$)$ What is the expected number of users able to continue using the system? Hint: Exploit

the property of the mean of a sum of r$.$ v$.\text{'}$s$.$