Suppose you had five sticks of length 1, 2, 3, 4, and 5 inches. If you chose three at random, what is the likelihood tht the three sticks could be put together, tip to tip, so as to form a triangle?
Now suppose you had twenty sticks, of lengths 1 through 20 inches. If you picked three at random, what is the likelihood that the three could be put together, tip to tip, to form a right triangle?
(Assume that a triangle has to have some area)
A viable triangle must have the longest side be greater than either of the other two sides but less than the sum of the other two sides. For the 5 stick scenario, there are 10 possible combinations ((5 x 4 x 3)/3! = 10). Of these only 3 (234,245, and 345) are valid. Therefore, the probability is 30%. For the 20 stick scenario there are 1,140 possible combinations (20 x 19 x 18)/3! According to a quick Excel spreadsheet approach there are 7 combinations that are right triangles (3, 4,5),(6,8,10), (5,12,13),(8,15,17),(9,12,15),(12,16,20) and (15,20,25). Therefore the odds of generating a right triangle out of 3 randomly selected integers between 1 and 20 inclusive would be 7/1,140 = 0.6%. Gordon S.
A Potential Solution
