document.write( "Question 1147178: Find how many arrangements there are of the letters of the word GUMTREE if the G is somewhere between the two Es? \n" ); document.write( "
Algebra.Com's Answer #768486 by Edwin McCravy(20055)\"\" \"About 
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document.write( "We can arrange the letters UMTR in 4! = 24 ways.\r\n" );
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document.write( "In each of those we will insert two E's, then we will insert a G somewhere\r\n" );
document.write( "between the two E's\r\n" );
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document.write( "Let XXXX represent any one of the 24 arrangements of UMTR.\r\n" );
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document.write( "Case 1. The E's are inserted together: \r\n" );
document.write( "EEXXXX, XEEXXX, XXEEXX, XXXEEX, XXXXEE\r\n" );
document.write( "For each of those 5 ways, there is only one way to insert the G between the two\r\n" );
document.write( "E's. So this case accounts for 5∙1 = 5 arrangements for each of the 24. \r\n" );
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document.write( "Case 2. The E's are inserted so that there is 1 letter between them: \r\n" );
document.write( "EXEXXX, XEXEXX, XXEXEX, XXXEXE. \r\n" );
document.write( "For each of those 4 ways, there are 2 ways to insert the G between the E's.\r\n" );
document.write( "[Those 2 ways are (1) just left of the letter between the two E's, and (2) \r\n" );
document.write( "just right of that letter between the two E's]\r\n" );
document.write( "So this case accounts for 4∙2 = 8 arrangements for each of the 24.   \r\n" );
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document.write( "Case 3. The E's are inserted so that there are 2 letters between them: \r\n" );
document.write( "EXXEXX, XEXXEX, XXEXXE.\r\n" );
document.write( "For each of those 3 ways there are 3 ways to insert the G between the 2 E's,\r\n" );
document.write( "So this case accounts for 3∙3 = 9 arrangements for each of the 24.\r\n" );
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document.write( "Case 4. The E's are inserted so that there are 3 letters between them: \r\n" );
document.write( "EXXXEX, XEXXXE.\r\n" );
document.write( "For each of those 2 ways there are 4 ways to insert the G between the 2 E's,\r\n" );
document.write( "So this case accounts for 2∙4 = 8 arrangements for each of the 24.\r\n" );
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document.write( "Case 5. The E's are inserted so that there are 4 letters between them: \r\n" );
document.write( "EXXXXE \r\n" );
document.write( "is the only way.\r\n" );
document.write( "There are 5 ways to insert the G between the two E's,\r\n" );
document.write( "So this case accounts for 1∙5 = 5 arrangements for each of the 24.\r\n" );
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document.write( "So for each of the 24 ways to arrange UMTR, there are 5∙1+4∙2+3∙3+2∙4+1∙5 =\r\n" );
document.write( "5+8+9+8+5 = 35 ways to insert two E's and a G somewhere between the two E's.\r\n" );
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document.write( "Therefore the answer is 24∙35 = 840 ways\r\n" );
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document.write( "Edwin
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