I am familiar with the standard proof that the binomial coefficient is an integer based on induction and Pascal's Rule.
I'm trying to prove that a related expression is an integer:
Using an adaptation of Pascal's Rule, we can see that and , but these equalities don't lead to an inductive proof.
Oh, yes, this is mathematically equivalent to the binomial coefficient, but I was hoping to find a more direct proof rather than converting to bionomial coefficient, and falling back on that standard proof.