SOLUTION: The number of scalene triangles having sides with integer lengths and perimeter less than 19 is: a)17, b)18, c)19, d)20, e)21 I have tried this over and over but the most that

Question 843979: The number of scalene triangles having sides with integer lengths and perimeter less than 19 is:
a)17, b)18, c)19, d)20, e)21
I have tried this over and over but the most that I can get is 12.
2,3,4
2,4,5
2,5,6
2,6,7
2,7,8
3,4,5
3,4,6
3,5,6
3,5,7
3,6,7
3,6,8
3,7,8
I know that any 2 sides must have a sum greater than the 3rd side.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I counted reasoning like this:

Perimeter=18:
The longest side must be less than
The possible side lengths, in decreasing order are:
8,7,3
8,6,4
7,6,5

Perimeter=17:
The longest side must be less than
The possible side lengths, in decreasing order are:
8,7,2
8,6,3
8,5,4
7,6,4

Perimeter=16:
The longest side must be less than
The possible side lengths, in decreasing order are:
7,6,3
7,5,4

Perimeter=15:
The longest side must be less than
The possible side lengths, in decreasing order are:
7,6,2
7,5,3

Perimeter=14:
The longest side must be less than
The possible side lengths, in decreasing order are:
6,5,3

Perimeter=13:
The longest side must be less than
The possible side lengths, in decreasing order are:
6,5,2
6,4,3

Perimeter=12:
The longest side must be less than
The possible side lengths, in decreasing order are:
5,4,3 (That's a right triangle).

Perimeter=11:
The longest side must be less than
The possible side lengths, in decreasing order are:
5,4,2

Perimeter=10:
The longest side must be less than
No possible solution, because with a longest side of length 4, the other two sides lengths could at most be 3 and 2, adding to a perimeter of .

Perimeter=9:
The longest side must be less than
The possible side lengths, in decreasing order are:
4,3,2

Perimeter=8:
The longest side must be less than
No possible solution, because with a longest side of length 3, we cannot have a scalene triangle with integer side lengths. Segments of lengths 3, 2, and 1 do not form a triangle because .
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