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A 10-cm stick has a mark at each centimeter. By breaking the stick at two of these nine marks at random, the stick is split into three pieces, each of integer length. What is the probability that the three lengths could be the three side lengths of a triangle? Express your answer as a common fraction.

 Nov 14, 2019
 #1
avatar+435 
+3

There are 36 ways to split the ruler, because you can do cuts at the 1st and 2nd notch, the 1st and 3rd notch, up to the 1st and 9th notch, then the 2nd and 3rd notch, up to the 2nd and 9th notch, the last one being the 8th and 9th notch, for a total of 8+7+...+1=36 ways.

There are only two combinations to make a triangle, 3-3-4 or 2-4-4, for a total of 6 permutations

6/36=1/6, so the required answer is seven.

 Nov 14, 2019
 #2
avatar+2499 
+1

There are only two combinations to make a triangle, 3-3-4 or 2-4-4, for a total of 6 permutations

 

 

The reason for this is because of the Triangle Inequality Theorem.

 

Meaning that the sum of the two legs of a triangle MUST be greater than the third side.

CalculatorUser  Nov 14, 2019
 #3
avatar+1681 
+1

The answer is also presented here:

 

 

https://brilliant.org/problems/from-signup/combinatorial-geometry-or-is-it-geometrical-combin/no-group/no-input/?topic_tag=combinatorics#

 Nov 14, 2019

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