document.write( "Question 943571: How many 5 letter arrangements can be made using the letters in the word INDEPENDENCE\r
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Algebra.Com's Answer #575337 by Edwin McCravy(20055)\"\" \"About 
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document.write( "INDEPENDENCE has 1 C, 2 D's, 4 E's, 1 I, 3 N's, and 1 P\r\n" );
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document.write( "There are 6 cases:\r\n" );
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document.write( "Case 1:  The 5 letter arrangements like the distinguishable\r\n" );
document.write( "         arrangements of VWWWW of which there are 5!/4! = 5.\r\n" );
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document.write( "Choose the W as E and the V any of the other 5 ways.\r\n" );
document.write( "That's 1*5 = 5 to multiply by 5.  So there are 25 arrangements for case 1.         \r\n" );
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document.write( "Case 2:  The 5 letter arrangements like the distinguishable\r\n" );
document.write( "         arrangements of VVWWW of which there are 5!/(2!3!) = 10.\r\n" );
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document.write( "We can only use D,E, and N in this case\r\n" );
document.write( "Choose the W 2 ways,as  E or N, and the V either of the remaining 2 ways.\r\n" );
document.write( "That's 2*2 = 4 to multiply by 10.  So there are 40 arrangements for case 2. \r\n" );
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document.write( "Case 3:  The 5 letter arrangements like the distinguishable\r\n" );
document.write( "         arrangements of VWXXX of which there are 5!/3! = 20.\r\n" );
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document.write( "We can choose the X 2 ways, as E or N, then the V and W as any combination\r\n" );
document.write( "of 2 letters from the remaining 5 letters, 5C2=10.\r\n" );
document.write( "That's 2*10 = 20 to multiply by 20.  So there are 400 arrangements for case 3. \r\n" );
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document.write( "Case 4:  The 5 letter arrangements like the distinguishable\r\n" );
document.write( "         arrangements of VWWXX of which there are 5!/(2!2!) = 30.\r\n" );
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document.write( "We can choose the W and X as any combination of 2 from the 3 letters D,E,N,\r\n" );
document.write( "3C2=3. Then we can choose the V as any one of the remaining 4 letters. \r\n" );
document.write( "That's 3*4 = 12 to multiply by 30. So there are 360 arrangements for case 4.\r\n" );
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document.write( "Case 5:  The 5 letter arrangements like the distinguishable\r\n" );
document.write( "         arrangements of VWXYY of which there are 5!/2! = 60.\r\n" );
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document.write( "We can choose the Y 3 ways, as D,E, or N. Then we can choose V, W and X\r\n" );
document.write( "as any combination of 3 from the remaining 5 letters, 5C3=10 \r\n" );
document.write( "That's 3*10 = 30 to multiply by 60. So there are 1800 arrangements for case 5.\r\n" );
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document.write( "Case 6:  The 5 letter arrangements like the distinguishable\r\n" );
document.write( "         arrangements of VWXYZ of which there are 5! = 120\r\n" );
document.write( "We can choose those as any combination of 5 from the 6 letters C,D,E,I,N,P.\r\n" );
document.write( "That's 6C5=6 to multiply by the 120.\r\n" );
document.write( "That's 6 to multiply by the 120.  So there are 720 arrangements for case 6.\r\n" );
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document.write( "Grand total for the 6 cases: 25+40+400+360+1800+720 = 3345 \r\n" );
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document.write( "Edwin
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