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| 5, 6, Pick Up Sticks (Posted on 2003-11-05) |
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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)
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Submitted by DJ
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Rating: 3.8750 (8 votes)
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Solution:
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3/10 (30%) and 1/190 (0.526%)
The number of ways you can pick up the three sticks is
5C3 = 5!/(5-3)!/3! = 5*4*3/6 = 10.
Of those ten ways of choosing the sticks, only three can make triangles (2-3-4, 2-4-5, and 3-4-5). This is because the sum of any two sides of a triangle must be greater than the other side.
Therefore, the probability of being able to form a triangle from the pieces is 3/10, or 30%.
For the second case, there are
20C3 = 20!/(20-3)!/3! = 20*19*18/6 = 1140
different ways to pick up the three sticks.
Of these, only six are right triangles (3-4-5, 6-8-10, 9-12-15, 12-16-20, 5-12-13, and 8-15-17).
Therefore, the probability of forming a right triangle is 6/1140 = 1/190, or around a .526% chance of forming a right triangle.
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