document.write( "Question 763864: In a polygon of 12 sides, how many diagonals are there? solve with the help of permutation & Combinations - Step wise? \n" ); document.write( "

Algebra.Com's Answer #465106 by ramkikk66(644) $\"\"$ $\"About$

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document.write( "A diagonal is a line that connects any 2 vertices of the polygon, right? \r\n" );
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document.write( "So, combinatorially, now many pairs of vertices can you choose in a 12 sided polygon?\r\n" );
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document.write( "Note that for a diagonal, the order of the vertices does not matter. \r\n" );
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document.write( "i.e. if a diagonal connects vertices A and B, it is the same as the \r\n" );
document.write( "diagonal connecting B to A and you would count it only once. \r\n" );
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document.write( "So the answer is \"12 choose 2\" or C(12,2) which is 12*11/1*2 = 66. \r\n" );
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document.write( "But here is the catch - this set of combinations would also include the \r\n" );
document.write( "12 edges of the polygon \r\n" );
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document.write( "i.e. if it had vertices from A to L, the edges AB, BC, CD etc. also would be counted in the above formula. \r\n" );
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document.write( "However, they are not diagonals but edges.\r\n" );
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document.write( "Hence we have to subtract 12 from the above answer to get the number of  diagonals.\r\n" );
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document.write( "In general, for a polygon of n vertices (n >= 3), the number of diagonals would be given by the formula\r\n" );
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document.write( "C(n,2) - n =  upon simplification.\r\n" );
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document.write( "Hope this helps. Do mail me if it's not clear.\r\n" );
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