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How many diagonals does a decagon have?

I have just learnt permutations, dispositions, combinations. How can I solve it with these concepts? I drew it and it was $35$ diagonals. How can I prove it with this method?

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A diagonal joins a vertex to one of the vertices that do not include that vertex itself and the immediately adjacent vertices. So: for each vertex there are seven diagonals. Times 10 equals 70; each diagonal is counted twice, so the final answer is 35.

Now, using combinations and such: There are $\binom{10}{2}\;$("10 choose 2") pairs of vertices, which equals 45. So there are 45 line segments joining pairs of vertices. Exactly 10 of those are sides of the decagon, the others are diagonals. Answer: 35. (Corrected; original had "10 choose 9" for no reason other than my lack of concentration.)

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  • $\begingroup$ I need a solutions using permutations, dispositions or combinations. $\endgroup$ – prishila May 24 '16 at 14:30
  • $\begingroup$ I realized that and just added it. $\endgroup$ – mathguy May 24 '16 at 14:32
  • $\begingroup$ why is it 10 choose 9 (or C10,9) $\endgroup$ – prishila May 24 '16 at 16:07
  • $\begingroup$ Combinations: In how many ways can you choose two distinct points out of a set of 10, when the order of the points doesn't matter? This is exactly what "combinations" means! $\endgroup$ – mathguy May 24 '16 at 16:08
  • $\begingroup$ I got it. We have C10,2 (10 choose 2)=10!/2!*8!=45. Then I will follow your reasoning. $\endgroup$ – prishila May 24 '16 at 16:15
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Formula for calculating number of diagonals of any polygon of n sides = n*(n - 3)/2

So here it's a decagon ,that is a 10 sided polygon, So n = 10. Simply plug value of n into the formula , you get: 10*(10-3)/2 = 35. Ans :)

(Note : no matter what sided polygon it is, you can find any no of diagonals in any polygon)

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  • $\begingroup$ What is your question? $\endgroup$ – bing Apr 15 '17 at 3:04
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(N-1 choose 2 ) -1

Example, (10-1 choose 2) -1 = (9 choose 2) -1 = 36-1 = 35 diagonal lines

This will work for any regular shape

Example 2/ 20 sides would be C19,2 -1 =170 Check: C20,2 - 20 =170

Very easy formula developed by Shawn Covrigaru

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