document.write( "Question 891657: in how many ways we can arrange the letters in the word 'MISSISIPI' so that all S's come together and no I's come together \n" ); document.write( "
Algebra.Com's Answer #539912 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I'll assume that MISSISSIPPI was mispelled. It has 4 S's and 2 P's.\r\n" );
document.write( "If you meant \"MISSISIPI\", there are so few you can just list them all\r\n" );
document.write( "easier than calculating them, for it's just the 6 permutations of\r\n" );
document.write( "M,P,(SSS) with 4 I's separating them:\r\n" );
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document.write( "IMIPISSSI\r\n" );
document.write( "IMISSSIPI\r\n" );
document.write( "IPIMISSSI     \r\n" );
document.write( "IPISSSIMI     \r\n" );
document.write( "ISSSIMIPI\r\n" );
document.write( "ISSSIPIMI\r\n" );
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document.write( "Assuming you meant the correct spelling of the state:\r\n" );
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document.write( "\"%22%22-%22%22\"\r\n" );
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document.write( "The number of ways the four S's can come together\r\n" );
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document.write( "The letters of MISSISSIPPI arranged in alphabetical order is\r\n" );
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document.write( "I,I,I,I,M,P,P,S,S,S,S\r\n" );
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document.write( "The number of ways the four S's can come together is the number of\r\n" );
document.write( "distinguishable permutations of these 8 things, where the (SSSS) is\r\n" );
document.write( "considered as a single \"thing\".\r\n" );
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document.write( "I,I,I,I,M,P,P,(SSSS)\r\n" );
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document.write( "Since there are 4 indistinguishable I's and 2 indistinguishable P's,\r\n" );
document.write( " the number is:\r\n" );
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document.write( "\"8%21%2F%284%212%21%29=840\"\r\n" );
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document.write( "From this 840 we must subtract the number of ways 2 or more I's \r\n" );
document.write( "can come together.\r\n" );
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document.write( "First we will calculate the number of distinguishable arrangements of\r\n" );
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document.write( "M,P,P,(SSSS) with no I's, and then we'll calculate how many ways we \r\n" );
document.write( "can insert the 4 I's with 2 or more together. Then we'll multiply\r\n" );
document.write( "those two numbers together.  \r\n" );
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document.write( "Again in the distinguishable arrangements of M,P,P,(SSSS), the \r\n" );
document.write( "(SSSS) is considered as just one thing.\r\n" );
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document.write( "Since there are 2 indistinguishable P's, the number is:\r\n" );
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document.write( "\"4%21%2F%282%21%29\" = 12\r\n" );
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document.write( "An example would be PMP(SSSS). Let's put a space before and after \r\n" );
document.write( "each letter or \"thing\" to indicate feasible places to insert the I's.\r\n" );
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document.write( "  _P_M_P_(SSSS)_\r\n" );
document.write( "            \r\n" );
document.write( "So there are 5 places to insert the four I's.\r\n" );
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document.write( "Case 1.  All four I's come together (IIII). \r\n" );
document.write( "         That's 5 ways.\r\n" );
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document.write( "Case 2.  The I's are split 3 and 1, (III) and (I).\r\n" );
document.write( "         Place the (III) any of 5 ways\r\n" );
document.write( "         Place the (I) in any of the 4 remaining places\r\n" );
document.write( "         That's 5*4 or 20 ways.\r\n" );
document.write( "Case 3.  The I's are split 2 and 2. (II) and (II)\r\n" );
document.write( "         Choose the 2 places for them to go in C(5,2) or 10 ways\r\n" );
document.write( "         That's 10 ways. \r\n" );
document.write( "Case 4.  The I's are split 2,1 and 1.  (II),(I), and (I)\r\n" );
document.write( "         Choose the places for the two single (I)'s C(5,2) = 10 ways\r\n" );
document.write( "         Choose the place for the (II) as any of the remaining 3 ways\r\n" );
document.write( "         That's 10*3 or 30 ways.\r\n" );
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document.write( "For those four cases that's 5+20+10+30 = 65 ways.\r\n" );
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document.write( "We multiply this 65 by the 12 distinguishable arrangements of M,P,P,(SSSS),  \r\n" );
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document.write( "That's 65*12 or 780 ways.\r\n" );
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document.write( "That's the number which we must subtract from the 840.\r\n" );
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document.write( "Final answer = 840 - 780 = 60 distinguishable permutations.\r\n" );
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document.write( "Here they all are, computer generated, 10 rows of 6 each:\r\n" );
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document.write( "IMIPIPISSSS    IMIPIPSSSSI   IMIPISSSSIP   IMIPISSSSPI   IMIPPISSSSI   IMIPSSSSIPI\r\n" );
document.write( "IMISSSSIPIP    IMISSSSIPPI   IMISSSSPIPI   IMPIPISSSSI   IMPISSSSIPI   IMSSSSIPIPI\r\n" );
document.write( "IPIMIPISSSS    IPIMIPSSSSI   IPIMISSSSIP   IPIMISSSSPI   IPIMPISSSSI   IPIMSSSSIPI\r\n" );
document.write( "IPIPIMISSSS    IPIPIMSSSSI   IPIPISSSSIM   IPIPISSSSMI   IPIPMISSSSI   IPIPSSSSIMI\r\n" );
document.write( "IPISSSSIMIP    IPISSSSIMPI   IPISSSSIPIM   IPISSSSIPMI   IPISSSSMIPI   IPISSSSPIMI\r\n" );
document.write( "IPMIPISSSSI    IPMISSSSIPI   IPPIMISSSSI   IPPISSSSIMI   IPSSSSIMIPI   IPSSSSIPIMI\r\n" );
document.write( "ISSSSIMIPIP    ISSSSIMIPPI   ISSSSIMPIPI   ISSSSIPIMIP   ISSSSIPIMPI   ISSSSIPIPIM\r\n" );
document.write( "ISSSSIPIPMI    ISSSSIPMIPI   ISSSSIPPIMI   ISSSSMIPIPI   ISSSSPIMIPI   ISSSSPIPIMI\r\n" );
document.write( "MIPIPISSSSI    MIPISSSSIPI   MISSSSIPIPI   PIMIPISSSSI   PIMISSSSIPI   PIPIMISSSSI\r\n" );
document.write( "PIPISSSSIMI    PISSSSIMIPI   PISSSSIPIMI   SSSSIMIPIPI   SSSSIPIMIPI   SSSSIPIPIMI\r\n" );
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document.write( "Edwin

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