How many passwords + rainbow-tables can there be?

Question

Mathematics

For example, I have already intercepted handshakes with the aircrack-ng utility (or rather airodump-ng).
I have a .cap file. I wondered how many password options there could be, but because I don’t know combinatorics well, I would like to make sure that I think correctly.
32 *2 (Cyrillic = lower and upper case) + 26*2 (Latin = lower and upper case) +

. , ! @ # $ % ^ & * ( ) _ - + = / " ' № : ? < > { } [ ]

28 (didn't forget anything?) + 10 (digits) = 154
So the options for an 8-digit password are 154^8 = 316348490636206336
Usually passwords make 8 - 16 characters.

c = sum([154**i for i in range(8,17)])
print(c)

It turns out: 100730461433216424107205973413148416. 36 digit number.
My video card can sort out, for example, 50,000 passwords / s.
In total it will take 2.0146092286643285e+30 seconds.
sec = c/50000.0
Or 5.5961367462898016e+26 hours or 2.331723644287417e+25 days (that's how long they live?)
Now the questions are: did I count everything correctly? Does it make sense to sort out passwords in this way?
There are also rainbow-tables. Do they speed things up?

Answer 1

I did not check the received numbers, but the message is correct.
Here you have answered your own question. :)
Brute force is a very reliable attack (it always works sooner or later), but an extremely inefficient attack. :)

Answer 2

1. What is a rainbow table? Once computed, it is reused by others for the "reverse hash" task.
2. Wi-Fi uses two parameters: ANonce and a password - so we have a challenge-response protocol designed specifically to make rainbow tables impractical - a hacking problem solved for one session will not help to hack another.
3. So how much to break? If you are trying to hack into someone else's network and put reasonable limits on someone else's password, you are. If you are setting up a security policy, it is better to take into account only the shortest of the options: if the limit is 8 ... 24 characters, then take exactly 8 for calculations.