Question 1199271
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All of the arrangements of the letters in the word ALGEBRA are written on a list, 
one of these arrangments selected at random. 
What is the probability that the arrangement contains a double-A
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<pre>
There are 7 letters in the word "ALGEBRA", in all; of them, only letter "A" repeats twice.


The table below shows the number of distinguishable arrangements with  1, 2, 3, 4, 5, 6 and 7 letters.
In the 2nd and 3rd columns, the formula is used nPk = n*(n-1)*...*(n-k+1).
In the second column, n=5.  In the third column, n=6.


      # of     # of arrangements  # of arrangements   # of arrangements 
     letters      without "A"       with one "A"        with two "A"s
     k  used         5Pk                6Pk                 {{{(k-1)!/2}}}
     --------------------------------------------------------------------

       1              5                  6                    -

       2             20                 30                    1

       3             60                120                    3

       4            120                360                   12

       5            120                720                   60

       6             -                 720                  360

       7             -                  -                  2520

     Total          325               1956                 2956


    The probability that the arrangement contains a double-A = {{{2956/(1+325+1956+2956)}}} = {{{2956/5238}}} = 0.5643  (rounded).


The "1", added in the denominator, is for empty arrangement.


<U>ANSWER</U>.  The probability that the arrangement contains a double-A is 0.5643  (rounded).
</pre>

Solved.


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The major lesson to learn from this solution is how to organize your thoughts and your calculations.