document.write( "Question 973681: How many ways can the letter in the word OPTIMISM be rearranged?\r
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\n" ); document.write( "2 Is, 2 Ms, 1 O, 1 P, 1 T, 1 S = (8,2)(6,2)(4,1)(3,1)(2,1)(1,1)\r
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\n" ); document.write( "\n" ); document.write( "which I got $\"+%288%21%2F%284%2A5%21%29%29+\"$ \r
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Algebra.Com's Answer #597276 by Edwin McCravy(20054) $\"\"$ $\"About$

You can put this solution on YOUR website!
How many ways can the letter in the word OPTIMISM be rearranged?\r
\n" ); document.write( "\n" ); document.write( "What I did:
\n" ); document.write( "2 Is, 2 Ms, 1 O, 1 P, 1 T, 1 S = (8,2)(6,2)(4,1)(3,1)(2,1)(1,1)
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\n" ); document.write( "which I got $\"+%288%21%2F%284%2A5%21%29%29+\"$
\n" ); document.write( "Did I do that correctly?
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document.write( "I'm afraid not.\r\n" );
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document.write( "OPTIMISM is an 8-letter word.  If the word were \"oPTimISM\" the answer would be\r\n" );
document.write( "8! =40320 because we would be able to tell the difference between the arrangements\r\n" );
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document.write( "mSiToPIM, mSIToPiM, MSIToPIm, MSiToPIm\r\n" );
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document.write( "But since we cannot because the I's and M's look the same, we must divide the\r\n" );
document.write( "9! by the number of ways the I's can be swapped, which is 2! and also by the\r\n" );
document.write( "number of ways the M's can be swapped, which is also 2!.\r\n" );
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document.write( "Answer =  = 10080\r\n" );
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document.write( "You see, we had to divide by 2!2! = 4 because, as we saw with the four arrangements\r\n" );
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document.write( "mSiToPIM, mSIToPiM, MSIToPIm, MSiToPIm, all look the same as MSITOPIM\r\n" );
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document.write( "every arrangement is counted 4 times too many in the 8!, that's why we divide\r\n" );
document.write( "by 4.  We divide by the factorial of the number of times each letter is\r\n" );
document.write( "repeated.\r\n" );
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document.write( "Edwin

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