The answer is 7,546,483,168,121,141. This assumes all possible skillbars with 8 skills (plus the empty bar). This does not take into account how the skills are ordered.
Here are a few other facts (also assuming only PvP skills)
- 270,006,654,759,760 of these bars have a resurrection signet in them. A randomly generated bar given a random primary and secondary (including blanks) would end up with a res sig in about 3.6% of them.
- There 188,735,881,177 possible skillbars consisting of assassin skills only (Res Sigs are too heavy to bring). The typical HvH Assassin utilizes less than five of these bars.
- There are 7,715,399,966,892 bars consisting of only skills from one profession. The ability to mix skills from another profession multiplies the number of possible combinations by 978.
- The number of possible GvG team compositions is approximately 10^127. This is around 50 orders of magnitude larger than the number of atoms in the observable universe.
- The amount of space required to store a wiki entry for every build is likely greater than the amount of space required to store every word uttered by every human being ever in existence, as a transcript of text.
Assumptions and restrictions made:
- Skill order is disregarded, including the type of skill itself. This includes lots of bars with assassin combos that can't be performed, and bars with seven glyphs.
- No skillbar can have more than one elite.
- Skillbars disregard primary and secondary profession. A Warrior/Necro, and Necro/Warrior with the same skillbar is counted as one.
Spreadsheet used to make calculations is attached.