document.write( "Question 830967: You have six sticks, lengths 1,2,3,4,5,6 inches, how many triangles can you make. Is there a specific formula for determining how many triangles can be made from different sets of segments in various lengths? \n" ); document.write( "
Algebra.Com's Answer #501117 by Edwin McCravy(20055)\"\" \"About 
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document.write( "The shortest side of a triangle must be longer than the difference\r\n" );
document.write( "between the other two sides.\r\n" );
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document.write( "Therefore the 1 inch stick cannot be used at all because it is not\r\n" );
document.write( "longer than the difference between any of the other sticks, for\r\n" );
document.write( "even if they differ by only 1, 1 is not longer than 1.\r\n" );
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document.write( "2 can be the shortest side if the other two sides differ by only 1.\r\n" );
document.write( "So we can have a triangle with (2,3,4), (2,4,5) or (2,5,6).\r\n" );
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document.write( "3 can be the shortest side if the other two sides differ by either \r\n" );
document.write( "1 or 2.  So we can have a triangle with (3,4,5), (3,4,6), or (3,5,6).\r\n" );
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document.write( "4 can only be the shortest side with (4,5,6)\r\n" );
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document.write( "So 7 triangles can be made with the 6 sticks 1,2,3,4,5,6.\r\n" );
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document.write( "Yes there are two formulas. One is for when there are an\r\n" );
document.write( "even numbers of sticks and the other is for when there \r\n" );
document.write( "are an odd number of sticks. \r\n" );
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document.write( "If you have sticks of lengths 1,2,3,...,n inches\r\n" );
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document.write( "The formula for when n is even is  n(n-2)(2n-5)/24\r\n" );
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document.write( "The formula for when n is odd is (n-1)(n-3)(2n-1)/24  \r\n" );
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document.write( "The case above is when there are 6 sticks. Since 6 is even,\r\n" );
document.write( "we use the formula\r\n" );
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document.write( "n(n-2)(2n-5)/24\r\n" );
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document.write( "6(6-2)(2·6-5)/24 = 6(4)(12-5)/24 = 6(4)(7)/24 = 7\r\n" );
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document.write( "Edwin
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