To guard against login attempts by bots, Alphabet Bank of India uses a captcha challenge on their net banking login page. The captcha text is a sequence of five letters from the English alphabet.
- An early version of the captcha text consisted of a sequence of five distinct letters picked uniformly at random from the lower-case English alphabet, $\{a, b, c, \dots, z\}$. Two examples of such captcha text are: $\textsf{qfhdk}$ and $\textsf{smgft}.$ An example of such captcha text which has its first two letters in increasing alphabetical order is: $\textsf{vxztr}.$ What is the probability that the captcha text generated in this way has the first two letters in increasing alphabetical order, and why?
- After a couple of years the bank noticed that some bots had become very good at cracking the captcha. So they updated their software so that it now generates the captcha text by picking five distinct letters uniformly at random from the English alphabet, where the case of each letter can be either upper or lower. That is, the letters are now picked from the set $\{a, b, c, \dots, z, A, B, C, \dots, Z\}$. Note that the same letter does not appear in both lower and upper case in the same captcha text. Two examples of the captcha text generated by this version are: $\textsf{QFhDk }$ and $\textsf{sMgfT.}$ Examples of such captcha text with the first two letters in increasing alphabetical order are: $\textsf{VxPTr}$ and $\textsf{vXptR}$. Note that the case of the letter does not matter when deciding the alphabetical order. What is the probability that the captcha text generated in this way has the first two letters in increasing alphabetical order, and why?
