Division of the Binomial Coefficient

Question

Prove that when p is prime, the binomial coefficient p!/(r!)((p-r)!) is divisible by p with r being greater than or equal to 1 and less than or equal to p-1 .

Clearly p! is divisible by p so I cant seem to see the need for any further working.

However the solution states that none of the prime factors of r! or p-r! are divisible by p (I do not disagree with this statement) whilst p! is evidently divisible by p. Hence p!/(r!)((p-r)!) is divisible by p.

Please could someone help explain the necessity of the prime factors explanation in the solution.

Answer 1

Consider the number $a=\frac{6}{3}$. Suppose someone told you

The numerator $6$ is divisible by $3$, so no need for further work: $a$ is divisible by $3$.

Where did they go wrong? Do you see why investigating the denominator is relevant?