document.write( "Question 1200281: Find the number of paths from A to B, moving up and right
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Algebra.Com's Answer #834424 by Edwin McCravy(20055) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Note (er, rather, rant) :)\r\n" );
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document.write( "Greenestamps was exactly right by his brute force solution. Below is the\r\n" );
document.write( "formal mathematical combinatorial solution. Brute force methods, especially\r\n" );
document.write( "computer brute force, have taken a lot of fun out of formal mathematics, as well\r\n" );
document.write( "as the need for even teaching it.  My observations are that the entire\r\n" );
document.write( "mathematics curriculum from high school up through the undergraduate college\r\n" );
document.write( "level was designed so that the student would be able to solve differential\r\n" );
document.write( "equations.   But nowadays, the computer can solve all differential equations by\r\n" );
document.write( "brute force.  I predict the day when no mathematics will be taught in schools.\r\n" );
document.write( "Why do math when you can get the answer by pointing and clicking with a mouse?\r\n" );
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document.write( "The figure below has the two omitted lines CD and EF in red. \r\n" );
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document.write( "First we'll find the number of paths for the entire grid including CD and EF.\r\n" );
document.write( "Then we'll subtract the number of paths that use one or both of CD and EF.\r\n" );
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document.write( "Let N(AB) = the number of paths from A to B each of which is described by a\r\n" );
document.write( "unique distinguishable permutation of RRRRRRRRRUUUU, 13 moves, 9 units right and\r\n" );
document.write( "4 units up, of which there are .\r\n" );
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document.write( "To find the number of paths to subtract from the 715, we will need the following\r\n" );
document.write( "partial paths:\r\n" );
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document.write( "Let N(AC) = the number of paths from A to C each of which is described by a\r\n" );
document.write( "unique distinguishable permutation of RRU, 3 moves, 2 units right and 1 unit up,\r\n" );
document.write( "of which there are .\r\n" );
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document.write( "Let N(DB) = the number of paths from D to B each of which is described by a\r\n" );
document.write( "unique distinguishable permutation of RRRRRRRUU, 9 moves, 7 units right and 2\r\n" );
document.write( "units up, of which there are .\r\n" );
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document.write( "Let N(AE) = the number of paths from A to E each of which is described by a\r\n" );
document.write( "unique distinguishable permutation of RRRRRUUU, 8 moves, 5 units right and 3\r\n" );
document.write( "units up, of which there are .\r\n" );
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document.write( "Let N(FB) = the number of paths from F to B each of which is described by a\r\n" );
document.write( "unique distinguishable permutation of RRRU, 4 moves, 3 units right and 1 unit\r\n" );
document.write( "up, of which there are .\r\n" );
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document.write( "Let N(DE) = the number of paths from D to E each of which is described by a\r\n" );
document.write( "unique distinguishable permutation of RRRU, 4 moves, 3 units right and 1 unit\r\n" );
document.write( "up, of which there are .\r\n" );
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document.write( "And of course, we need, the number of very short partial paths, of 1 move each. \r\n" );
document.write( "N(CD) = 1 and N(EF) = 1,\r\n" );
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document.write( "The answer will be\r\n" );
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document.write( "N(AB) - [N(AC)*N(CD)*N(DB) + N(AE)*N(EF)*N(FB)] + [N(AC)*N(CD)*N(DE)*N(EF)*N(FB)]\r\n" );
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document.write( "The middle term counts TWICE the paths through both CD and EF, and thus\r\n" );
document.write( "subtracts that number twice.  This is why we need the third term, to correct by\r\n" );
document.write( "adding on the number of paths through both CD and EF. Calculating,\r\n" );
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document.write( "715 - [3*1*36 + 56*1*4] + [3*1*4*1*4] = 431.\r\n" );
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document.write( "Edwin
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