How many diagonals does a decagon have? 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?
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.)
(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