document.write( "Question 1178350: Find the number of ways of arranging the letters in the word DOMINATE if
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\n" ); document.write( "c) the odd-numbered positions must be vowels.
\n" ); document.write( "d) the last two letters must be T and E.(
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Algebra.Com's Answer #807752 by Edwin McCravy(20060)\"\" \"About 
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Find the number of ways of arranging the letters in the word DOMINATE if
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document.write( "We can choose the 1st letter in any of 8 ways.\r\n" );
document.write( "We can choose the 2nd letter in any of 7 ways.\r\n" );
document.write( "We can choose the 3rd letter in any of 6 ways.\r\n" );
document.write( "We can choose the 4th letter in any of 5 ways.\r\n" );
document.write( "We can choose the 5th letter in any of 4 ways.\r\n" );
document.write( "We can choose the 6th letter in any of 3 ways.\r\n" );
document.write( "We can choose the 7th letter either of 2 ways.\r\n" );
document.write( "We can choose the 8th letter only 1 way.\r\n" );
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document.write( "Answer = (8)(7)(6)(5)(4)(3)(2)(1) = 8! = 40320.

\n" ); document.write( "b) the first letter must be a vowel.
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document.write( "We can choose the 1st letter in any of 4 ways, any one of {O,I,A,E}\r\n" );
document.write( "We can choose the 2nd letter in any of 7 ways.\r\n" );
document.write( "We can choose the 3rd letter in any of 6 ways.\r\n" );
document.write( "We can choose the 4th letter in any of 5 ways.\r\n" );
document.write( "We can choose the 5th letter in any of 4 ways.\r\n" );
document.write( "We can choose the 6th letter in any of 3 ways.\r\n" );
document.write( "We can choose the 7th letter either of 2 ways.\r\n" );
document.write( "We can choose the 8th letter only 1 way.\r\n" );
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document.write( "Answer = (4)(7)(6)(5)(4)(3)(2)(1) = (4)(7!) = (4)(5040) = 20160.

\n" ); document.write( "c) the odd-numbered positions must be vowels.
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document.write( "We can choose the 1st letter in any of 4 ways, any one of {O,I,A,E}\r\n" );
document.write( "We can choose the 3rd letter in any of 3 ways.\r\n" );
document.write( "We can choose the 5th letter in either of 2 ways.\r\n" );
document.write( "We can choose the 7th letter in only 1 way.\r\n" );
document.write( "We can choose the 2nd letter in any of 4 ways.\r\n" );
document.write( "We can choose the 4th letter in any of 3 ways.\r\n" );
document.write( "We can choose the 6th letter either of 2 ways.\r\n" );
document.write( "We can choose the 8th letter only 1 way.\r\n" );
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document.write( "Answer = (4)(3)(2)(1)(4)(3)(2)(1) = (4!)(4!) = (24)(24) = 576.

\n" ); document.write( "d) the last two letters must be T and E.
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document.write( "We can choose the 7th letter only 1 way, as T.\r\n" );
document.write( "We can choose the 8th letter only 1 way, as E.\r\n" );
document.write( "We can choose the 1st letter in any of 6 ways.\r\n" );
document.write( "We can choose the 2nd letter in any of 5 ways.\r\n" );
document.write( "We can choose the 3rd letter in any of 4 ways.\r\n" );
document.write( "We can choose the 4th letter in any of 3 ways.\r\n" );
document.write( "We can choose the 5th letter in either of 2 ways.\r\n" );
document.write( "We can choose the 6th letter in only 1 way.\r\n" );
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document.write( "Answer = (1)(1)(6)(5)(4)(3)(2)(1) = 6! = 720.\r\n" );
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document.write( "[Note: I am assuming that the last two letters must be \"TE\" and not \"ET\".\r\n" );
document.write( "It's a little unclear on that point.  If it could end in \"ET\" the answer\r\n" );
document.write( "would be twice as much.]\r\n" );
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document.write( "Edwin
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