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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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closed as off-topic by Weapon of Choice, studiosus, Ivo Terek, anorton, hardmath Jul 21 at 3:19

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3 Answers 3

up vote 0 down vote accepted

$(26C4*10C1+26C3*10C2+26C2*10C3)*5!$

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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

HINT:

  • 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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Hint.

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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