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A valid password is a 5 character string made up of letters (A,B, . . . , Z) and numbers (0, 1, . . . 9) such that at least one number and at least two letters are used. How many valid passwords are there?

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It works better if you give us some idea of why you are interested in the problem, what you know about it, what you've tried, where you get stuck, and so on. –  Gerry Myerson Jan 29 '13 at 4:01
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3 Answers

up vote 0 down vote accepted


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quick question. does the 5! at the end represent the different ways of permuting the way the 5 characters can be arranged in a password? –  dotdotdot Jan 30 '13 at 9:08
Yes it does..... –  Abhra Abir Kundu Jan 30 '13 at 10:44
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A valid password can be of the following type.

  1. 4 letters, 1 number.
  2. 3 letters 2 numbers
  3. 2 letters and 3 numbers.

So count number of possibilities of each type and sum them up.

It will be good if you write your (partial) solution.

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  • How many $5$-character strings are possible altogether?

  • How many of them are invalid because they contain no digit?

  • How many of them are invalid because they contain no letter? How many are invalid because they contain exactly one letter?

  • Are there any that are invalid because they contain too few digits and too few letters?

Now put the answers together to get the result.

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