A box contains 7 sticks of the following lengths 3cm, 4cm, 5cm, 6cm, 7 cm, 8cm, 9cm. three sticks are drawn at random from the box. What is the probability that can form a triangle with those sticks?

• 
  345,346,456,457,458,467,468,469,567,568,569,578,579,678,679
  total 15 triplet
  probability=15/7c3
  =3/7
• 10 years agoHelpfull: Yes(1) No(6)
• 345,346,456,457,458,467,468,469,478,479,489,567,568,569,578,579,589,678,679,689,789 total triplet=21
  probability=21/7c3
  
• 9 years agoHelpfull: Yes(1) No(4)
• Ans:
  25
  42
  
  
  Triangle inequality theorem states that sum of the lengths of any two sides of a triangle must be greater than the third side.
  
  Suppose a,b,c are the sides of a triangle. Then
  a + b > c
  a + c > b
  b + c > a
  
  But we don't need to check all these conditions. Just add the shorter sides and compare with the longer side.
  If sum of the shorter sides is greater than the longer side, triangle can be formed. Else, triangle cannot be formed.
  
  With this information, we can approach the given problem.
  
  Total ways in which 3 numbers can be selected from (2, 3, 4, 5, 6, 7, 8, 9,10) = 9C3 = 84
  
  Using the triangle inequality theorem, we can see that the following combinations cannot form a triangle.
  
  (2,3,5), (2,3,6), (2,3,7), (2,3,8), (2,3,9), (2,3,10)
  (2,4,6), (2,4,7), (2,4,8), (2,4,9), (2,4,10)
  (2,5,7), (2,5,8), (2,5,9), (2,5,10)
  (2,6,8), (2,6,9), (2,6,10)
  (2,7,9), (2,7,10)
  (2,8,10)
  
  (3,4,7), (3,4,8), (3,4,9), (3,4,10),
  (3,5,8), (3,5,9), (3,5,10),
  (3,6,9), (3,6,10)
  (3,7,10)
  
  (4,5,9),(4,5,10)
  (4,6,10)
  
  i.e., 34 combinations cannot form triangle out of the total 84
  
  P(selected sides cannot form a triangle)
  =
  34
  84
  
  
  P(selected sides can form a triangle) =
  1
  −
  34
  84
  =
  50
  84
  =
  25
  42
• 5 years agoHelpfull: Yes(0) No(0)
• Answer is 35/11.
  
  Hint: "The sum of two numbers should be greater than the third side."
  
  So, choose the combinations based on hint will result in 11 combinations. And the probability of choosing 3 sticks from 7 sticks is 7C3 = 35.
  
  Total Probability = 35/11.
• 5 years agoHelpfull: Yes(0) No(0)