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The lengths of six non-collinear line-segments are 3, 4, 5, 6, 7 and 8 units. The maximum number of scalene triangles that can be formed by using these line segments is
All lengths are distinct, so every valid triangle is scalene, and there are $$\binom{6}{3} = 20$$ ways to choose three lengths. A choice fails if the two smaller sides do not exceed the largest, which happens only for $$(3, 4, 7)$$, $$(3, 4, 8)$$ and $$(3, 5, 8)$$. So the number of triangles is $$20 - 3 = 17$$.
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