Question 1202209
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A password must be created using 4 letters from the word "PASSWORD". 
In how many ways can the {{{highlight(cross(letter))}}} <U>letters</U> be arranged?
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<pre>
In this word "PASSWORD", there are 7 different letters; one letter "S" is repeated twice.


Let's consider first the case, when all 4 letters of a password are different.
There are  7*6*5*4 = 840 such passwords (arranging).


Now consider the opposite case, when two letters in password are "S".
There are  {{{C[6]^2}}} = {{{(6*5)/2}}} = 3*5 = 15 ways to select the remaining 
two letters from 6 = 8-2 letters, different from "S".


Further, this number 15 we should multiply by the number of different arranging of 4 letters,
two of which are identical "S".


So the number of these passwords with two "S" is  {{{15*(4!/2!)}}} = 15*12 = 180.


In all, we have the number of all possible distinguishable passwords/arranging  840 + 180 = 1020.    <U>ANSWER</U>
</pre>

Solved.



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<H3>My note to note by @greenestamps</H3>


In his post-solution note, tutor @greenestamps writes


<pre>
    NOTE: The solution shown by tutor @ikleyn is very likely the question that the problem INTENDED to ask.

    However, the wording of the problem implies that 4 letters are chosen from the letters in PASSWORD; and then THOSE 4 LETTERS are arranged.
</pre>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The wording of the problem &nbsp;DOES &nbsp;NOT &nbsp;implies that &nbsp;4 &nbsp;letters are just chosen.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;To assert this is to pervert the problem.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;He solves the problem in his interpretation; me - in my interpretation.



The fact that two different interpretations are possible means 
a crash, a catastrophe and a death sentence for this problem, 
as it is written in the post.



........................



Obviously, the person who created this problem, is unprofessional &nbsp;Math composer 
and creates his compositions &nbsp;" kneeling in the garage ", even without reading any Math textbooks
and learning Math exclusively from reading this and other Internet web-sites.


A correctly worded and the &nbsp;ONLY &nbsp;POSSIBLE &nbsp;question should be &nbsp;" how many different passwords are possible ? "