You have 6 sticks of lengths 10, 20, 30, 40, 50, and 60

Question

You have 6 sticks of lengths 10, 20, 30, 40, 50, and 60 centimeters. The number of non-congruent triangles that can be formed by choosing three of the sticks to make the sides is

A. 3
B. 6
C. 7
D. 10
E. 12

OA will be posted after some time.

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Bunuel · Accepted Answer

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

Based on this there can be only 7 triangles formed: (20, 30, 40), (20, 40, 50), (20, 50, 60), (30, 40, 50), (30, 40, 60), (30, 50, 60), (40, 50, 60).

Answer; C.

premnath · Answer

Following method may be applied-

As Bunuel has mentioned

"The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides."

This a very important characteristic of triangle.

Now coming to this particular problem, we have sides mentioned as 10, 20, 30, 40, 50, 60.
Difference between any tow sides is 10, therefore 10 cannot be any side of a triangle. 
So, we are left with 20, 30, 40, 50 and 60. ->> 5 sides.

Now, Number of triangles can be formed with 5 sides is  ^^5C {^}3   = 10 
However, following triangles are not possible-
   -- as 20 + 30 (NOT) > 50

Similarly     is not possible
Similarly     is not possible

Easiest way to eliminate triangles is to start with least two sides and sum them up. All the sides having equal or higher that value are eliminated. In our case we started with 20 and 30. Hence, 50, and 60 are eliminated.
Repeat the same process with the next number along with the least one. In our case 20 and 40. So, 40 is eliminated.   ]

So, finally the number of triangles is  10-3 =7

Hope, this is more logical and quicker method of solving such problem.

fameatop · Answer

Hi Bunuel,

Is there any other method (combinatorics) to solve this question ?

b9n920 · Answer

Got this correct :-D

but spent 3.48

Used trial and error, keeping the basic properties of triangle in mind. Anybody has a better / faster method?? Thanks.

Fdambro294 · Answer

1st, you can not form a valid triangle with any other 2 sides if 1 of the Sides picked =10. The Triangle will always fail the Triangle Test for Validity (the 2 Smaller Side Lengths will NEVER be greater than > Longest Side Length)

The 7 Triangles that will work after using the TEST: 2 Shorter Side Lengths > 3rd Longest Side Length

20 - 30 - 40

20 - 40 - 50

20 - 50 - 60

30 - 40 - 50

30 - 40 - 60

30 - 50 - 60

40 - 50 - 60

Since there are only "5 Choose 3" = 20 Different Combinations of 3 Sticks Picked, if you can quickly realize that 10 can never be a valid Side for Any Triangle formed with any of the 2 other sides given, you can cut that down to 10 Triangle Values that you need to Test.

Moving quickly, it's possible to get it finished within 2 - 3 minutes.

EDIT: I said it's "possible. I didn't say I was able to do it lol.

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You have 6 sticks of lengths 10, 20, 30, 40, 50, and 60

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You have 6 sticks of lengths 10, 20, 30, 40, 50, and 60

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