15,834 Possible outcomes from taking the quiz
Luckily for you, someone (by the name of Kira Starthestral) already reversed engineered the wand quiz, and compiled the results into a nice spreadsheet. (Note that you have to look through ALL the tabs, which contain complete information. The first is only a selection of some wand surveys.) Pretty much this entire answer is either copied directly from that spreadsheet or was made based on its data. (You can also use my pdf if you find that easier.)
It is possible that the actual quiz was modified for the new Pottermore, (just like the Sorting Quiz was). However, seeing as the questions have not changed, this is probably unlikely.
It is even more likely that I messed up on my math somewhere.
- 38 woods
- 21 lengths
- 14 flexibilities
- 6 colors
- 3 cores
- 3 shapes
Which would give a total of 603288 theoretical combinations. (The other answer arrived at 19152 by miscounting the number of flexibilities and by ignoring shape and color.)
We can lower that upper bound even further by looking at the answer choices:
- Height - (3 choices)
- Eye color - (9 choices)
- Birthday date - (2 choices)
- Trait most proud of - (7 choices)
- Path - (3 choices)
- Fear - (5 choices)
- Artefact - (7 choices)
Putting the bound down at 39690, Showing that some combinations are impossible to get.
However, this is also not good, because some wands have multiple possible ways to reach.
Going back to our list of results,
The length is determined by the artefact you choose from the trunk and your height.
The core is determined by your greatest fear and the artefact you choose from the trunk.
The flexibility is determined by the trait you're most proud of and the date of your birthday.
The wood is determined by your eye colour, the trait you're most proud of and the path you choose.
The wood determines the wand's colour.
The shape of the wands is determined by the flexibility.
Length has a unique output for each of it's 21 inputs.
Flexibilty/shape has a unique output for each of its 14 inputs.
Core and wood are more tricky as they overlap with other questions (making not all results possible) and have less outputs then inputs (making numeorus duplicates.)
I calculated cores and wood as having 13/5 and 145/7 possible results per wand respectfully. (No idea what the proper terminology for this is.)
Thus, there are a total of 15,834 (21*14*13/5*145/7) possible wand results to get from the quiz.
39690 possible answer combinations
603288 possible theoretical possible wands (including shape & color)
33516 possible theoretical possible wands (not including shape & color)
15834 possible different quiz results
17682 wands within the bounds of results but impossible to arrive at based on the current structure of the quiz.
Basically, if you are trying to get a specific characters wand, you probably won't be able to. You could get Harry's or Ron's, but others can be impossible by either requiring results outside the bounds of the quiz (e.g. mahogany), having wands impossible to reach, or having wands that are impossible to reach while maintaining the other info that we know of the characters (e.g. eyecolor).
Note: Due to (what is probably) a bug in the new Pottermore, there are currently only thirteen possible flexibilities, resulting in just 14,703 possible results.