Exercise 1.53 (Password problem)

Question

Fred needs to choose a password for a certain website. Assume that he will choose an 8-character password, and that the legal characters are the lowercase letters a, b, c, . . . , z, the uppercase letters A, B, C, . . . , Z, and the numbers 0, 1, . . . , 9.
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(a) How many possibilities are there if he is required to have at least one lowercase letter
in his password?
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(b) How many possibilities are there if he is required to have at least one lowercase
letter and at least one uppercase letter in his password?
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(c) How many possibilities are there if he is required to have at least one lowercase
letter, at least one uppercase letter, and at least one number in his password?

fifthist x · Accepted Answer

\begin{enumerate}[label=(\alph*)]

\item $62^{8} - 36^{8}$

\item $62^{8} - 36^{8} - 36^{8} + 10^{8}$

\item $62^{8} - 36^{8} - 36^{8} - 10^{8} + 2(62^{8} - 36^{8} - 52^{8} + 26^{8}) 
+ 62^{8} - 36^{8} - 36^{8} + 10^{8}$

\end{enumerate}

Sher · Answer

The solution to part C by fifthist x is wrong. An easy way to check it is that it sums up to greater than 62^8, which shouldn’t be the case since that is the total number of possibilities.  
We can get to the right answer by using three properties:

1. P(A) = 1 - P(Acomplement)

2. DeMorgan’s Law: 
AnB = AcomplementUBcomplement

3: Inclusion Exclusion formula

Exercise 1.53 (Password problem)

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Exercise 1.53 (Password problem)

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