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A certain security system has passwords that consist of 6 characters. The passwords are case sensitive and must satisfy the following requirements:

  • Contain 4 letters (from 26 letters "a" to "z"), repeats are allowed
  • At least one letter must be lowercase
  • At least one letter must be uppercase
  • Contain 2 digits (from 10 digits "0" to "9")

How many different passwords are possible in this system?

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  • $\begingroup$ What are your thoughts on the problem? $\endgroup$ Oct 8 '20 at 2:20
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Just implement the multiplication rule several times.

First, choose 2 positions to hold the digits. Then, enumerate the possible arrangements of these two digits. Third, do the same with the four positions left for letters. Finally, carefully filter out the schemes where all letters are in uppercase or lowercase.

The answer is $\displaystyle\binom62\times10^2\times26^4\times(2^4-2)$.

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