Latest self Aptitude Question SOLUTION: If you have sticks which measure 1, 2, 3, 4, 5, 6, 7, 8 and 9 cm (one of each) how many different triangles could you make using three of the sticks

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If you have sticks which measure 1, 2, 3, 4, 5, 6, 7, 8 and 9 cm (one of each) how many different triangles could you make using three of the sticks

Vignesh (12 years ago)

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Pargat Pal Singh  (12 years ago)

I am getting 34.

you can not make triangle with 1,2,3 sticks.

In a triangle , sum of two sides will always be greater than third side.

considering this , I could get 34 triangles only.

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Atul  (12 years ago)

Dear Rakesh,

I think only 34 triangles can be made.

Pls check that (1,3,5), (1,3,6),(1,3,7),(1,3,8), (1,3,9) and (2,3,6)........(3,4,8),(3,4,9),(3,5,9) are also not possible.

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Dr.Dipin Singh  (12 years ago)

As calculated by Rakesh, 84 combinations are possible, but in some cases triangles can not be made.

Such cases are 50.

I shall try to give the list of such cases .
123,124,125,126,127,128,129.......7 cases
134,135,136,137,138,139 ..........6 cases
145,146,147,148,149...............5 case
156,157,158,159 ..................4 case
167,168,169 ...................... 3 case
178,179........................... 2 cases
189................................. 1 case

then 235,236,237,238,239..............5 cases
246,247,248,249.......................4 case
257,258,259........................... 3 case
268,269 .............................. 2 case
279 .................................. 1 case

next 347,348,349,358,359,369 ........... 6 case
also 459 ............................... 1 case

total 50 cases are not possible.

only 84-50=34 cases are possible.

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Rakesh Kumar Kohli  (12 years ago)

We know that in a triangle , sum of two sides will always be greater than third side. So following combination of sticks cannot make triangles.
(1,2,3),(1,3,4),(1,4,5),(1,5,6),(1,6,7),(1,7,8),(1,8,9),
(2,3,5),(2,4,6),(2,5,7),(2,6,8),(2,7,9),
(3,4,7),(3,5,8),(3,6,9),
(4,5,9)
So 16 combinations cannot make triangles.
Three sticks can be choosen from the given 9 sticks in 9C3 = 84 ways. Out of these 84 combinations 16 combinations cannot make triangles. So only 68 triangles can be made by using these sticks.

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