How many ways can we make triangles?

You cannot take any three lengths and form a triangle with them.
There are some triangles that won't work.
The basic rule about the lengths of triangles is:
A<B+C
B<A+C
C<A+B
If you want to see more about this check out my
visual explanation.

There are many many ways to think about this problem.  One of the biggest problems is that it is very easy to get bogged down in the many combinations.  Because of this, it is very important to be careful about repetition.

My Special Approach
I found it was very useful to distinguish between A, B, and C by establishing that
C>B>A
This is not a truth, I am just saying this for the purposes of  solving this problem.
To find out all the ways to make a triangle I will first pick a value for C, anywhere from 3-50 (lower bound is 3 because C must always be greater than two other triangles).  Then pick a value for B.  The first condition for B is that it can be no more than C-1.  These selections are for purposes counting.  When I pick, for example, C=50 I am then going to look at each case where B=49, 48, 47, .....,3, 2.  This will get very large since we will have to do the same thing for every C, but eventually we will turn it into a sum of a sum and let the computer take care of it.
For each pair C and B (where C>B) we can determine for what values of A we can make a triangle and (more importantly) how many values for A are possible:
If C>B and B>A then there will be 2B-1-C different values that we could pick for A.  
Click here to see the proof for this.

If you pick a C value and look at what happens for different B values you will find that there are certain restrictions to what B can be.
To be specific, if C is even:         B can be no less than C/2+1
and if C is odd: B can be no less than (C+1)/2
To see why this is true, down load this excel spreadsheet containing a visual explanation in chart form.
At this point, I'm pretty much ready to show my formula for computing the number of ways to make a triangle.