Prove that for any prime $p$, if $a>b$ then $p^2$ divides $C(pa,pb)-C(a,b)$.

Question

Let, $p$ be a prime and $a>b$. If $\operatorname{C}(n,r)$ denotes the combination of $r$ objects from a collection of $n$ objects taken at a time, prove that $\operatorname{C}(pa,pb)-\operatorname{C}(a,b)$ is divisible by $p^2$.

Tried using De Polignac's formula, but, it is getting difficult and laborious and it isn't working. Then I tried to fix $b$ and apply induction on $a$. It is also getting extremely difficult to handle the calculations arising from it. How can I attack this problem now? Because just breaking them down and writing explicitly is not a good option I guess.

Answer 1

This was proved by Charles Babbage in 1819. The statement for divisbility by $p^3$ is known as Wolstenholme's theorem. Also see that page (or this answer and one of its comments) for a combinatorial proof of the divisibility by $p^2$.