document.write( "Question 1026495: how many distinguishable permutations can be formed from the letters in the word success?\r
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Algebra.Com's Answer #641788 by ikleyn(52778) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "how many distinguishable permutations can be formed from the letters in the word success?\r
\n" ); document.write( "\n" ); document.write( "-----This is the first time I am seeing a problem like this. I'm sure it is simple to understand, I just need a quick explanation.
\n" ); document.write( "Thanks!
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document.write( "1. Let us start from more simple case. Imagine that you have the word consisting of three identical letters, \r\n" );
document.write( "   for example, the word \"ooo\" of three \"o\".\r\n" );
document.write( "   How many distinguishable permutations can you have with three \"o\"?\r\n" );
document.write( "   Think one minute. But of course, only one. Nothing more.\r\n" );
document.write( "   You can move these letters from place to place, but you can not get nothing new.\r\n" );
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document.write( "2. Next, imagine that you have the word \"ogo\".\r\n" );
document.write( "   How many permutations can you have?\r\n" );
document.write( "   Only three: \"goo\", \"ogo\" and \"oog\". Nothing more.\r\n" );
document.write( "   The theory says that you can have 6 permutations of three objects.\r\n" );
document.write( "   But it is only in the case when all three your objects are distinguishable (which is assumed by default).\r\n" );
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document.write( "3. OK. So, you see that there is a problem here.\r\n" );
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document.write( "4. Now consider the word \"Ogo\" of three letters, consisting of one capital \"O\", \"g\" and small (lover case) \"o\".\r\n" );
document.write( "   Let us assume that we are able to make a distinguish between capital \"O\" and lover-case \"o\". (But of course, we can).\r\n" );
document.write( "   How many permutations do we have of these three symbols? Here they are in two lines:\r\n" );
document.write( "       gOo, Ogo, Oog,\r\n" );
document.write( "       goO, ogO, oOg.\r\n" );
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document.write( "   Until we can distinguish \"O\" and \"o\", we have 6 permutations.\r\n" );
document.write( "   But once we loss this ability, we have only three of them.\r\n" );
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document.write( "   Why? Simply because every two permutations (every two 3-letter words) that differ in the ordering the letters \"O\" and \"o\" \r\n" );
document.write( "   in the line (in the string) merge into ONE 3-letter word. They TWO simply become ONE 3-letter word. \r\n" );
document.write( "   TWO initially different permutations/words become ONE word.\r\n" );
document.write( "   EVERY two initially different permutations that differ in the ordering \"O\" and \"o\" merge and become ONE permutation.\r\n" );
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document.write( "5. This analysis works for any n-letter word that contains two identical letters and (n-2) different remaining letters.\r\n" );
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document.write( "   Therefore, instead of n! permutations of n distinguished objects you will have only half of them \r\n" );
document.write( "   in the case when TWO objects are identical (undistinguished). The half is .\r\n" );
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document.write( "6. If you followed me to this point, you can easily continue on your own and consider the case when your n-letter word \r\n" );
document.write( "   contains TWO different pairs of undistinguished letters.\r\n" );
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document.write( "   How many permutations will you have in this case?\r\n" );
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document.write( "   .\r\n" );
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document.write( "7. Now apply it to the word \"success\".\r\n" );
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document.write( "   Ooo! It contains THREE \"s\" and TWO \"c\".\r\n" );
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document.write( "   So, you need to make one more step further.\r\n" );
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document.write( "   My answer is  .\r\n" );
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document.write( "   2!  goes for two \"c\" and  3!  goes for three \"s\".\r\n" );
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document.write( "   And yours ??\r\n" );
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