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Math Help - Combinatorics

  1. #1
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    Combinatorics

    An octagon has 8 sides. How many diagonals does an octagon have?

    Little confused with this question. . can anyone help?

    Thanks
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    Senior Member vincisonfire's Avatar
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    Let's take one vertex. It can be linked to 7 other vertices.
    Let's take the next vertex. It can also be linked to 7 other vertices but there is one diagonal that is already drawn. Therefore, there is 6 new diagonals.
    ...
     \sum_{i=1}^7 i = \frac{i(i+1)}{2} = 28 *
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  3. #3
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    Quote Originally Posted by vincisonfire View Post
    Let's take one vertex. It can be linked to 7 other vertices.
    Let's take the next vertex. It can also be linked to 7 other vertices but there is one diagonal that is already drawn. Therefore, there is 6 new diagonals.  \sum_{i=1}^7 i = \frac{i(i+1)}{2} = 28 *
    Unfortunately, the answer given above is an over count.
    The answer is 20. An octagon is a subgraph of complete graph, K_8 on eight vertices. K_8 has 28 edges but only 20 are diagonals of the octagon.
    In general, for an n sided figure there are {n \choose 2}-n = \frac {n(n-1)}{2}-n = \frac {n(n-3)}{2} diagonals.
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    Senior Member vincisonfire's Avatar
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    My apologies. Counting sides for diagonals
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  5. #5
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    Quote Originally Posted by Carrick
    What did it mean by diagonals? I'm doing simple combinations right now, your theory seems a little heavy

    The above quote is from a PM: please ask such questions in the open. Others may also benefit from the answer.

    An octagon is a figure with eight vertices and eight edges. If two edges share a vertex, then the edges are said to be adjacent. But any two vertices will determine a line segment. If you join each vertex with each other vertex we have what is known as a complete graph. In the case of eight vertices the complete graph, K_8, has 28 edges. The diagonals are the edges that are in the original octagon. Thus 20 are diagonals.

    Thus in K_n\;,\;n\ge 3 there are {n \choose 2} = \frac {n(n-1)}{2} edges.
    But n of the edges are not diagonals.
    Attached Thumbnails Attached Thumbnails Combinatorics-octogon.gif  
    Last edited by Plato; December 2nd 2008 at 03:06 PM.
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