The Enigma machine was used in World War II to send encrypted messages. There were five

rotors to choose from. Three rotors were chosen and placed in order in the machine, and each was

set to one of $26$ letters $($A$-$Z$)$ to create the starting position. $($There were also other complications

such as a plugboard which swapped ten pairs of letters, but we will ignore this here.$)$ Suppose

you are a cryptanalyst trying to break Enigma.

$($a$)$ How many possible starting positions are there?

$($b$)$ Now suppose you know which three rotors are being used $($and the order$),$ but you don$$t

know the letters on each one that form the setting. Find the probability that:

$($i$)$ the setting is DKS

$($ii$)$ the setting contains $2$ vowels

$($iii$)$ the setting begins and ends with a consonant

$($iv$)$ the setting contains at least one A

$($c$)$ Repeat the calculations in $($b$)$ if no two rotors can have the same letter.