document.write( "Question 848964: In class we considered the rearrangement of the word 'ignominious'.
\n" ); document.write( "a) How many rearrangements are possible if the first letter must be an 'i' and the last letter must be an 's'?
\n" ); document.write( "b) How many rearrangements are possible if the first letter must be a constant? (Be careful - the calculations are not the same for all constants)
\n" ); document.write( "c) How many rearrangements are possible if the vowels must always be separated by atleast one constant?
\n" ); document.write( "d) How many rearrangements are possible if the constants must be kept together? \n" ); document.write( "
Algebra.Com's Answer #512039 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "First learn that the letters that are not vowels are called \"consonants\",\r\n" );
document.write( "not \"constants\".\r\n" );
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document.write( "\"ignominious\" has 6 vowels and 5 consonants.  There are \r\n" );
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document.write( "3 indistinguishable i's \r\n" );
document.write( "2 indistinguishable o's\r\n" );
document.write( "2 indistinguishable n's \r\n" );
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\n" ); document.write( "In class we considered the rearrangement of the word 'ignominious'.
\n" ); document.write( "a) How many rearrangements are possible if the first letter must be an 'i' and the last letter must be an 's'?
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document.write( "i _ _ _ _ _ _ _ _ _ s\r\n" );
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document.write( "That's the number of distinguishable arrangements of the 9 other\r\n" );
document.write( "letters \"gnominiou\", which has 2 indistinguishable n's, 2 indistinguishable\r\n" );
document.write( "o's and 2 indistinguishable i's.\r\n" );
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document.write( "Answer: \"9%21%2F%282%212%212%21%29\" = 45360\r\n" );
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\n" ); document.write( "b) How many rearrangements are possible if the first letter must be a consonant? (Be careful - the calculations are not the same for all consonants)
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document.write( "Case 1:  g comes first. g _ _ _ _ _ _ _ _ _ _ _ \r\n" );
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document.write( "That's the number of distinguishable arrangements of the 10 other\r\n" );
document.write( "letters \"inominious\", which has 3 indistinguishable i's, 2 indistinguishable\r\n" );
document.write( "n's and 2 indistinguishable o's.\r\n" );
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document.write( "\"10%21%2F%283%212%212%21%29\" = 151200 ways\r\n" );
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document.write( "Case 2:  n comes first. n _ _ _ _ _ _ _ _ _ _ _ \r\n" );
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document.write( "That's the number of distinguishable arrangements of the 10 other\r\n" );
document.write( "letters \"igominious\", which has 3 indistinguishable i's, and 2 indistinguishable o's\r\n" );
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document.write( "\"10%21%2F%283%212%21%29\" = 302400 ways\r\n" );
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document.write( "Case 3. m comes first.  Same as case 1, which was 151200\r\n" );
document.write( "Case 4. s comes first.  Same as case 1, which was 151200\r\n" );
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document.write( "3×151200 + 302400 = 756000\r\n" );
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\n" ); document.write( "c) How many rearrangements are possible if the vowels must always be separated by atleast one consonant?
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document.write( "Since we have 6 vowels and only 5 consonants, the only \r\n" );
document.write( "configuration possible is to alternate them like this:\r\n" );
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document.write( "VCVCVCVCVCV\r\n" );
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document.write( "We can place the 6 vowels distinguishably \"6%21%2F%283%212%21%29\" ways,\r\n" );
document.write( "since the 3 i's are indistinguishable and the 2 o's are \r\n" );
document.write( "indistinguishable.\r\n" );
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document.write( "For each way we can place the 6 vowels, we can place the 5\r\n" );
document.write( "consonants distinguishably in \"5%21%2F%282%21%29\" ways since the 2 n's \r\n" );
document.write( "are indistinguishable.\r\n" );
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document.write( "So that's \"6%21%2F%283%212%21%29\"\"%22%D7%22\"\"5%21%2F%282%21%29\" = 60×60 = 3600\r\n" );
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\n" ); document.write( "d) How many rearrangements are possible if the consonants must be kept together?
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document.write( "There are 7 ways the 5 consonants can come together\r\n" );
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document.write( "1. CCCCCVVVVVV\r\n" );
document.write( "2. VCCCCCVVVVV\r\n" );
document.write( "3. VVCCCCCVVVV\r\n" );
document.write( "4. VVVCCCCCVVV\r\n" );
document.write( "5. VVVVCCCCCVV\r\n" );
document.write( "6. VVVVVCCCCCV\r\n" );
document.write( "7. VVVVVVCCCCC\r\n" );
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document.write( "Since the number of ways to place the 11 letters distinguishably \r\n" );
document.write( "in any given configuration of constants and vowels is the same as in\r\n" );
document.write( "any other given configuration, we can just use the 3600 ways from part\r\n" );
document.write( "(c) and multiply it by 7.\r\n" );
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document.write( "Answer 3600×7 = 25200\r\n" );
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document.write( "Edwin
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