How many hexagons can be constructed by joining the vertices of a 15 sided polygon if none of the sides of the hexagon is also the side of the 15-gon.
Suppose, we take a 9 sided polygon and use Gap Method (Gap and string method in permutations and combinations) to fill other 6 points to be filled in between the present 9 points in order to have a 15 sided polygon
Also, in order to make hexagon according to the question, we just have to choose 6 "gaps" between the present points and put the points in the places. So, by doing all this we will have a 15 sided polygon.
And in this way, no. of ways of making a 15 sided polygon will mean same as selecting 6 points from a 15 sided polygon which aren't adjacent. (which helps in the idea of making a hexagon which will have no side which is also a side of the 15 sided polygon)
[ "×" - present points, "_" - gaps]
× _ × _ _ ×× _ _ ×× _ _×× _ _×No. of gaps=9
So, now according to me,
No. of such Hexagons which can be formed = $\binom{9}{6}$= 84
But, this is not the correct answer. What am I missing?
(Correct Answer- How many hexagons can be constructed by joining the vertices of a 15 sided polygon if none of the sides of the hexagon is also the side of the 15-gon.)
