document.write( "Question 946748: How many five-letter words are there in which no letter appears 3 or more times in a row?
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Algebra.Com's Answer #578056 by Edwin McCravy(20055) $\"\"$ $\"About$
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document.write( "Indirectly:\r\n" );
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document.write( "There are 26⁵ ways to arrange 5 alphabet letters in a row\r\n" );
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document.write( "That's 11881376 ways\r\n" );
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document.write( "We need to subtract the unacceptab;le ones:\r\n" );
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document.write( "1. Those which have exactly 5 letters in a row all the same\r\n" );
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document.write( "12345\r\n" );
document.write( "XXXXX\r\n" );
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document.write( "There are 26 ways to choose what X represents.\r\n" );
document.write( "That's 26 ways\r\n" );
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document.write( "That's 26 ways to subtract\r\n" );
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document.write( "2. Those which have exactly 4 letters in a row the same:\r\n" );
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document.write( "12345  12345  \r\n" );
document.write( "XXXX-  -XXXX\r\n" );
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document.write( "Choose the position for the four in a row in 2 ways\r\n" );
document.write( "Choose the X 26 ways\r\n" );
document.write( "Choose the - 25 ways\r\n" );
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document.write( "That's 2x26x25 = 1300 more ways to subtract\r\n" );
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document.write( "3. Those which have exactly 3 letters in a row on either end:\r\n" );
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document.write( "12345  12345\r\n" );
document.write( "XXX--  --XXX \r\n" );
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document.write( "Choose the position for the three in a row in 2 ways\r\n" );
document.write( "Choose the X 26 ways\r\n" );
document.write( "Choose the - that's next to an X in 25 ways\r\n" );
document.write( "Choose the - on the end in 26 ways (it can be the same as X or its neighbor)\r\n" );
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document.write( "That's 2x26x25x26 = 33800 ways to subtract\r\n" );
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document.write( "4. Those which have exactly 3 letters in a row in the middle:\r\n" );
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document.write( "12345\r\n" );
document.write( "-XXX- \r\n" );
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document.write( "Choose the X 26 ways\r\n" );
document.write( "Choose the left - in 25 ways\r\n" );
document.write( "Choose the right - in 25 ways \r\n" );
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document.write( "That's 26x25x25 = 16250 more ways to subtract\r\n" );
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document.write( "Total = 11881376 - 26 - 1300 - 33800 - 16250 = 11830000\r\n" );
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document.write( "------------------------------\r\n" );
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document.write( "Direct.\r\n" );
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document.write( "1.  Those which have no two in a row:\r\n" );
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document.write( "12345\r\n" );
document.write( "-----\r\n" );
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document.write( "Choose letter #1 26 ways\r\n" );
document.write( "Choose letter #2 25 ways\r\n" );
document.write( "Choose letter #3 25 ways (can be the same as 1 but not 2\r\n" );
document.write( "Choose letter #4 25 ways (can be the same as 1 or 2 but not 3\r\n" );
document.write( "Choose letter #5 25 ways (can be the same as 1 or 2 or 3 but not 4  \r\n" );
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document.write( "That's 26x25x25x25x25 = 10156250\r\n" );
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document.write( "2.  Those which have exactly 1 pair of letters in a row.\r\n" );
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document.write( "(a) 12345   (b) 12345   (c) 12345   (d) 12345\r\n" );
document.write( "    XX---       -XX--       --XX-       ---XX\r\n" );
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document.write( "Choose the position for the 1 pair in a row in 4 ways\r\n" );
document.write( "Choose the letter for X in 26 ways\r\n" );
document.write( "Choose letter #3 in (a), letter #4 in (b), letter #2 in (c) \r\n" );
document.write( "and letter #3 in (d) in 25 ways\r\n" );
document.write( "Choose letter #4 in (a), letter #5 in (b), letter #1 in (c) \r\n" );
document.write( "and letter #2 in (d) in 25 ways\r\n" );
document.write( "Choose letter #5 in (a), letter #1 in (b), letter #5 in (c) \r\n" );
document.write( "and letter #1 in (d) in 25 ways\r\n" );
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document.write( "That's 4x26x25x25x25 = 1625000\r\n" );
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document.write( "3.  Those which have exactly 2 pairs of letters in a row\r\n" );
document.write( "on either end.\r\n" );
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document.write( "12345  12345  \r\n" );
document.write( "XXYY-   XXYY \r\n" );
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document.write( "Choose the position for the 2 pair in a row in 2 ways\r\n" );
document.write( "Choose the letter for X in 26 ways\r\n" );
document.write( "Choose the letter for Y in 25 ways\r\n" );
document.write( "Choose the - in 25 ways\r\n" );
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document.write( "That's 2x26x25x25 = 32500\r\n" );
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document.write( "4.  Those which have two pairs on each end:\r\n" );
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document.write( "12345   \r\n" );
document.write( "XX-YY   \r\n" );
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document.write( "Choose the letter for X in 26 ways\r\n" );
document.write( "Choose the - in 25 ways\r\n" );
document.write( "Choose the letter for Y in 25 ways\r\n" );
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document.write( "That's 26x25x25 = 16250\r\n" );
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document.write( "Total = 10156250 + 1625000 + 32500 + 16250 =  11830000\r\n" );
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document.write( "Edwin
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