Who is asking: Parent
Level: Secondary
Question:
Consider a 30 sided polygon. If three diagonal are selected at random,
what is the PROBABILITY that they share a common endpoint?
The solution is (405)(52)(25)/405 choose 3
I just dont see how this is possible!
Hi,
First one must note that a diagonal here does not include an edge of the polygon. Thus the number of diagonals is 30C2 -30 = 405.
The number of ways to pick three diagonals in thus 405C3.
To find the number of ways to get three diagonals through a point we first choose a point in 30C1 = 30 ways and then pick the other three end points of these diagonals; this can be done in 27C3 ways (you can't pick points adjacent to the one that you picked already as that forms an edge of the polygon, not a diagonal).
The probability that three diagonals selected at random go through a common point is thus
p = [30(27C3)]/405C3 = 30*27*26*25/405*404*403 = 2*26*25/404*403
Denis