Algebra.Com's Answer #838739 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "The letters of the word MATHEMATICS are written, one on each of 11 separate cards.
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document.write( "The cards are laid out in a line.
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document.write( "(a) Calculate the number of different arrangements of these letters.
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document.write( "(b) Determine the probability that the vowels are placed together.
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document.write( "(a) The word MATHEMATICS has 11 letter.\r\n" );
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document.write( " Of them, letters M, A, T are repeating; the other 8 = 11-3 letters are unique.\r\n" );
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document.write( " Letter M has multiplicity 2; the same with letters A and T.\r\n" );
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document.write( " Therefore, the number of different arrangements of the letters of word MATHEMATICS is\r\n" );
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document.write( "  =  = 4,989,600. ANSWER\r\n" );
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document.write( "(b) The block of vowels is AEAI. It consists of 4 letters.\r\n" );
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document.write( " The other 11-4 = 7 letters are consonants.\r\n" );
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document.write( " When we consider arrangements, the block of vowels can be placed in any of 7+1 = 8 \r\n" );
document.write( " possible positions between consonants, or before consonants, or after consonants.\r\n" );
document.write( " It gives 8 possible options.\r\n" );
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document.write( " Next, there are 4! = 24 possible permutations inside this block,\r\n" );
document.write( " and there are 7! possible permutations of consonants outside of this block.\r\n" );
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document.write( " So, the total number of such permutations for word MATHEMATICS is the product\r\n" );
document.write( " 8*4!*7!\r\n" );
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document.write( " To get the number of possible arrangements, we still (or again) must divide it by the factor (2!*2!*2!).\r\n" );
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document.write( " So, the number of all possible arrangements of this type (b) is \r\n" );
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document.write( "  =  = 120,960. \r\n" );
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document.write( " The desired probability is the ratio of numbers 120,960 and 4,989,600\r\n" );
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document.write( " P =  =  = 0.02424 (rounded). ANSWER\r\n" );
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document.write( "Solved.\r
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