The chance to find the solution is not some fixed value, the difficulty changes every 2016 blocks (approx. 2 weeks) depending on how fast those 2016 blocks were solved.
It's not exactly mathematics, it's just calculating the result of cryptographic hash function. And they don't guess the full hash (which is by the way isn't "64 digits", but simply 256 bits, which are often represented as 64 hexadecimal characters, but this is simply an encoding), they only need to find a hash with certain amount of zeros at the start, this amount is actually the difficulty of mining.
And you don't need some secret software for it, software is available for free, what you need is specialized hardware. Also, luck - this thing doesn't exist. Miners don't rely on luck, they join their forces together and share the reward between themselves proportionally to their contribution.
Only a 4 digit number you have 10,000 possible combinations and for the 64 digits I don’t think I should even start writing it .
For example if you are solving a problem which would need a random 4 digit number and you can repeat then , you have 10 choices for all the digits , 10*10*10*10 , really hard to express but yeah , its really a hard job to do .
Miners don't use "permutation and combination", they just feed the block data and nonce to the SHA-256 function, and they chance nonce until they get a hash value with the needed amount of leading zeros.
1. You do agree that it is represented as a hexadecimal number so do not argue about that , I am quite Familiar with the facts you stated but my main idea was to give a mathematical side to the problem . Which could have been done only by hexadecimal digit consideration , which is VALID.
2. They need leading zeroes because the repetitions are allowed , practically needing any value in between 0-9 is actually valid .
3.Plus the difficulty changes to maintain the neutrality of the situation ' if there are many people struggling to get the number it will become more and more difficult with time therefore they have to adjust it every 2 weeks.
4. Plus read before answering * I specifically wrote , one would need permutation and combination if they try and solve it on paper*
If you would have read the paragraph you would have realized that whole idea was to break it down in terms of permutation and combinations which is the main ancestors of the function they use today .
I do think I stated amazing *computer software machine* I was wrong in the sense I did not know you would try and run the heavy hardware with a lame software. I do believe that they need to have all things considered and equipped .
Plus if you are using any free software from the web I do think you realize that most of them come with a lot of viruses therefore I do believe that they also need a good software with a good machinery and computer.
Hope that answers your accusations.
Have a nice day