Binomial coefficient
Mostly minor things, but could still be helpful:


Functions should generally be named in a verb form, so rename the function to something like calculate_binomial_coefficient().
This is a little hard to read:


if (r_mult / mult != r)



and could use more parenthesis for clarity:

if ((r_mult / mult) != r)

Consider using curly braces for these single-line conditional statements.  It could also help with avoiding some additional bugs, if you end up adding anything to them.
Since this is C++ and not C, use std::uint64_t.  Try to use all the equivalents that are found in the std namespace.