There are   =  = 10 ways to choose 3 distinct numbers of given 5 numbers.


It is not astronomically large number, and I can list all 10 combinations in the list  below:


 1)  1 2 3    (3 = 1 + 2, degenerate)

 2)  1 2 4    (4 > 1 + 2, impossible)

 3)  1 2 5    (5 > 1 + 2, impossible)

 4)  1 3 4     (4 = 1 + 3, degenerate)

 5)  1 3 5     (5 > 1 + 3, impossible)

 6)  1 4 5     (5 = 1 + 4, degenerate)

 7)  2 3 4

 8)  2 3 5     (5 = 2 + 3, degenerate)

 9)  2 4 5

10)  3 4 5 


In the next column on the right, I commented each combination/configuration.


"degenerate" means degenerated triangle.
"impossible" means a case where the triangle inequality is violated.
Empty comment means "a triangle is possible".


From the table, there are only 3 possible triangles ## 7, 9 and 10.    ANSWER