Can you solve this matchstick triangle problem?

I found this problem on YahooAnswers. Given 24 matchsticks, in how many ways can they be arranged to form a single triangle? Two triangles are considered the same if they are congruent. There is a way of attacking this systematically that will give the answer with just a bit of effort.

For those who would like to take it to the next level, see if you can find an approximation formula for n matches. I found a very simple formula that gives the exact value for special cases that can be arbitrary large. In the other cases, the percent error, though not the error itself, seems to keep getting smaller as n increases. I checked the formula against a program I wrote that generalizes the method of solution that can be used for the 24 matchstick problem. In some cases, increasing the number of matches results in fewer triangles. You can see an example of this for n=11 compared to n=12.