Q$3.$ What is the probability of these events when we randomly select a permutation of the $26$ lowercase letters of the English alphabet?

a$)$ The first $13$ letters of the permutation are in alphabetical order.

b$)a$ is the first letter of the permutation and $z$ is the last letter.

c$)$ a and $z$ are next to each other in the permutation.

d$)$ a and $b$ are not next to each other in the permutation.

e$)$ a and $z$ are separated by at least $23$ letters in the permutation.

f $)z$ precedes both a and $b$ in the permutation.