SOLUTION: How many scalene triangle exist whose sides a,b,c are natural numbers less than 8?

Question 832226: How many scalene triangle exist whose sides a,b,c are natural numbers less than 8?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Since the sides are all different numbers,
and the numbers to choose from are 1, 2, 3, 4, 5, 6, and 7,
your choices are limited.
You can make 35 sets of 3 different numbers,
but not all sets will be possible sides of a scalene triangle,
because the longest side must be longer than the sum of the lengths of the other two sides.
Otherwise you cannot make a triangle.

I see no choice but to make a list.
With longest side measuring 7,
the other side lengths (in decreasing order) could be
1) 6 and 5,
2) 6 and 4,
3) 6 and 3,
4) 6 and 2,
5) 5 and 4, or
6) 5 and 3.
If the longest sides measured 7 and 4,
you would need a smaller third side length that added to 4 gives a sum of more than 7,
and there in no natural number that works for that.
So we cannot make any triangles answers to this problem whose longest sides are 7 and 4.
The same goes for 7 and 3, or 7 and 2.
If the longest side measures 7, our options are only the 6 choices listed above.

With the longest side measuring 6,
the other side lengths (in decreasing order) could be
1) 5 and 4,
2) 5 and 3,
3) 5 and 2, or
4) 4 and 3.

With the longest side measuring 5,
the other side lengths (in decreasing order) could be
1) 4 and 3, or
2) 4 and 2.

With the longest side measuring 4,
the other side lengths (in decreasing order) could only be
1) 3 and 2.

I see no other choices that would work.
Segments with lengths of 3, 2, and 1 do not make a triangle.
If I did not miscount the possible choices, there are only
possible triangles.
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