Consider this function:
F(n, p) = ⁿC_p - ⌊n/p⌋
where n is a positive integer and p is a prime number.

Is F(n, p) always divisible by p?
If so, prove it.
If not, provide a counterexample.

Notes: ⁿC_p is the number of combinations of n elements taken p at a time. It is also known as Binomial Coefficient and read as "n choose p".
• ⌊m⌋ is equal to floor of m which is the greatest integer less than or equal to m