So there's a small game where when we see a four digit number somewhere in public, my friends and I would try to form the value of ten using these four digits with only basic operations.
If we see a four-digit number: $5829$
a possible way is to do:
$$9 + 5 - 8 \div 2 = 10$$
When I say basic I mean operations only include "$+,\div,-,\times"$ and the digits can be used in any order as done above.
Are there any sufficient conditions to guarantee that a string of numbers can attain a value of $10$? How many solutions are there?