Question 1192625
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How many different 4-letter permutations can be formed from the letters in the word WESTINGHOUSE?
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            It is clear that the correct question is to find the number of all distinguishable  4-letter arrangements
            (not permutations).


            So,  I will solve the problem in this {{{highlight(fixed)}}} formulation.



<pre>
The given word consists of 12 letters. 

Of them, 8 letters are non-repeating; two letters (E and S) are repeating and have multiplicity 2.


So, I first consider 10 unique letters  W, E, S, T, I, N, G, H, O, U and will calculate the number of 4-letter 
arrangements of these letters.  All these arrangements consist of non-repeating letters.


Then I will consider all 4-letter distinguishable words having two repeating "E" and no repeating "S";

then I will consider all 4-letter distinguishable words having two repeating "S"  and no repeating "E";

then I will consider all 4-letter distinguishable words having two repeating "E" and two repeating "S".


At the end, I will add all these opportunities.



(1)  the number of 4-letter arrangements of 10 distinct letters W, E, S, T, I, N, G, H, O, U  is

        10*9*8*7 = 5040.



(2)  the number of all 4-letter distinguishable words having two repeating  "E"  is 

         {{{C[4]^2*9*8}}} = {{{((4*3)/2)*9*8}}} = 6*9*8 = 432.

         Here {{{C[4]^2}}} = {{{(4*3)/2}}} = 6  is the number ow ways to select two positions for E in the 4-letter word,
         without looking the order.  Next, the factors 9 and 8 are to calculate the number of placing the rest unique letters
         at the remaining two positions in the 4-letter word. 



(3)  the number of all 4-letter distinguishable words having two repeating  "S"  is 

         {{{C[4]^2*9*8}}} = {{{((4*3)/2)*9*8}}} = 6*9*8 = 432.

         Here {{{C[4]^2}}} = {{{(4*3)/2}}} = 6  is the number ow ways to select two positions for S in the 4-letter word,
         without looking the order.  Next, the factors 9 and 8 are to calculate the number of placing the rest unique letters
         at the remaining two positions in the 4-letter word. 



(4)  the number of 4-letter distinguishable words having two repeating "E" and two repeating "S" is

         {{{4!/(2!*2!)}}} = {{{24/4}}} = 6.



Finally, I sum up the found numbers  5040 + 432 + 432 + 6 = 5910.


<U>ANSWER</U>.  There are  5910 such distinguishable 4-letter words.
</pre>

Solved.