document.write( "Question 1192625: How many different 4-letter permutations can be formed from the letters in the word WESTINGHOUSE? \n" ); document.write( "

Algebra.Com's Answer #824522 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "How many different 4-letter permutations can be formed from the letters in the word WESTINGHOUSE?
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\n" ); document.write( "\n" ); document.write( "            It is clear that the correct question is to find the number of all distinguishable 4-letter arrangements
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\n" ); document.write( "\n" ); document.write( "            So, I will solve the problem in this $\"highlight%28fixed%29\"$ formulation.\r
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document.write( "The given word consists of 12 letters. \r\n" );
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document.write( "Of them, 8 letters are non-repeating; two letters (E and S) are repeating and have multiplicity 2.\r\n" );
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document.write( "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 \r\n" );
document.write( "arrangements of these letters.  All these arrangements consist of non-repeating letters.\r\n" );
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document.write( "Then I will consider all 4-letter distinguishable words having two repeating \"E\" and no repeating \"S\";\r\n" );
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document.write( "then I will consider all 4-letter distinguishable words having two repeating \"S\"  and no repeating \"E\";\r\n" );
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document.write( "then I will consider all 4-letter distinguishable words having two repeating \"E\" and two repeating \"S\".\r\n" );
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document.write( "At the end, I will add all these opportunities.\r\n" );
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document.write( "(1)  the number of 4-letter arrangements of 10 distinct letters W, E, S, T, I, N, G, H, O, U  is\r\n" );
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document.write( "        10*9*8*7 = 5040.\r\n" );
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document.write( "(2)  the number of all 4-letter distinguishable words having two repeating  \"E\"  is \r\n" );
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document.write( "          =  = 6*9*8 = 432.\r\n" );
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document.write( "         Here  =  = 6  is the number ow ways to select two positions for E in the 4-letter word,\r\n" );
document.write( "         without looking the order.  Next, the factors 9 and 8 are to calculate the number of placing the rest unique letters\r\n" );
document.write( "         at the remaining two positions in the 4-letter word. \r\n" );
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document.write( "(3)  the number of all 4-letter distinguishable words having two repeating  \"S\"  is \r\n" );
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document.write( "          =  = 6*9*8 = 432.\r\n" );
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document.write( "         Here  =  = 6  is the number ow ways to select two positions for S in the 4-letter word,\r\n" );
document.write( "         without looking the order.  Next, the factors 9 and 8 are to calculate the number of placing the rest unique letters\r\n" );
document.write( "         at the remaining two positions in the 4-letter word. \r\n" );
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document.write( "(4)  the number of 4-letter distinguishable words having two repeating \"E\" and two repeating \"S\" is\r\n" );
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document.write( "          =  = 6.\r\n" );
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document.write( "Finally, I sum up the found numbers  5040 + 432 + 432 + 6 = 5910.\r\n" );
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document.write( "ANSWER.  There are  5910 such distinguishable 4-letter words.\r\n" );
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