Ridiculously rare - **1/64300886496** or **1.555188512 x 10^-11** (roughly) or **0.000000001555188512%** chance. Put it simply, you have more chance of winning a lottery with the whole world (yes all 7 billionish people) competing than getting a Pokemon with Perfect IVs. So if you can get one go try your luck in a lottery ASAP.

My calculations are alot more miniscule but i'll outline my logic on them beforehand;

You have 192 values in total (6 x 32). There are 6 values plucked out of random out of these, and your aim is to have them all at the max. So basically every time you pluck 6 random numbers out of this, a new combination is formed, and the order of the numbers in the combination does not matter, as there are 6 different categories. 6/192 is the chance of getting a SINGLE perfect IV (6 perfect IV counts in 192 numbers) on any stat - in which that would be 3.1% like the pervious answer.

So if you want perfect IVs on every stat - there are 64300886496 possible combinations

So the chance is **1/64300886496**.

There is a formula for finding the combination which I might put here later :P

(n!)/[(n-r)!(r!)]

Also, I may have interpreted your question wrong. I'm assuming you want all perfect IVs, since the question would probably say something like 'How rare is it for a Pokemon to have a perfect IV?' rather than a 'Perfectly IVed Pokemon?'. Anyway sorry if I interpreted your question wrong :P. Also might have done the calculations wrong :s DT double check for me! :P

Also - note this only applies to meeting a Pokemon with max IV's. Breeding, I'm not sure about, since the IVs are influenced depending on the parents.