What is the fastest way to estimate the number of pieces in a large collection?
What a tricky (bricky!) question. 
Let's estimate...

By Weight

So, can one estimate the number of bricks in a bag or box by weight?

In fact, they do weigh bags and boxes over at the factory to make sure there aren't any parts missing. But of course they're just comparing expected weights. 


That's why you can't simply weigh a big mix of unknown bricks to
calculate their number. There's no reference.


However, you can use a scale to estimate a number of bricks, if you have sorted them out individually. 

If you keep your pieces in trays or drawers and you have a big bin of identical pieces (2x4's, 2x1's, ...), you can estimate them by weighing them. 


Weigh them all, then divide by the weight of a single one.


Gets close. Speeds things up. 



By Numbers

Anyway, try to narrow things down. To do so, collect as much data as possible. 


Which sets do you own?
Look up the number of pieces per set. 
Take into account if you bought an incomplete set. How much is missing?
Did you mailorder pieces? 
Can you estimate the amount? Maybe check your orders/ accounts.
Did you buy a considerable number of bulk?


All this should already give you a rough number, considering the sets/mailorder/bulk ratio. 



By Eye


Do you sort out your Lego into trays and drawers? You could count some of them, calculate an average amount and project it.
What about the relevance of certain parts? Do they count? Where there's lego technic, there's plenty of technic pins. You could simply leave out those piles of pins and separators or
weigh them. If they're sorted out, of course.




(Disclaimer: Do not take this last part too seriously...)

Or is it all in ONE BIG BIN?

Well, let's see. 


How can one estimate the number of people in a crowd?


Take an aerial view, divide it into sections, count people in one frame and multiply by frames. As long as people are spread out evenly, you should get close enough. 



Example: There are (maybe) 63 minifigs in this picture. Count all minifigs in one of the frames and multiply them by four for a rough answer. 




How can one estimate the number of jellybeans in a jar?




Find out about jar and jellybeans, diameter, height, volume. Calculate. As long as you're not getting peckish, you should win big.



Now, how can one estimate the number of bricks of completely different shapes and sizes in ONE BIG BIN?


In theory, you would stirr everything to get one humongeous mass.
That is, you want bricks, pins, wheels, minifigs, separators all next
to each other, not in different layers with small parts at the bottom
of the bin. Next, pour it out and spread it evenly. Take a photo. Use the method
mentioned above.
Or, leave it in the bin. Take out one cup full of Lego,
count the parts, determine how many cups your bin holds and
extrapolate.


Conclusion: You're better off counting. While sorting out.