If a password is made by choosing three letters from A to F, what is the probability of the letters being in alphabetical order?

Example Answer

If the password uses three distinct letters from A through F, with all ordered passwords equally likely, the probability is 1/6. There are 6 times 5 times 4, or 120, possible passwords. Any chosen set of three letters has six possible orders, and exactly one is alphabetical. Equivalently, C(6, 3) gives 20 alphabetically ordered passwords, so the probability is 20/120.

I would check whether letters may repeat and whether “alphabetical” means strictly increasing. If repetition is allowed but the letters must strictly increase, the numerator stays 20 and the denominator becomes 6 cubed, giving 5/54. If repeated letters such as AAB count as alphabetical, we instead count nondecreasing sequences.

Worked example

With repetition allowed and nondecreasing order, there are C(8, 3) = 56 valid sequences out of 216, giving 7/27.

Why this works

The answer does not divide an unordered selection count by another unordered count when the outcome is an ordered password.

Follow-up to prepare for

Ask about replacement and the meaning of alphabetical order before calculating.