The product of $n$ consecutive integers is divisible by $n!$, using induction in conjunction with $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$

Question

I have seen this proof which uses induction. It seems to be a way of proving the property $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$, instead of using this in the induction.
I have tried to start with the binomial property and then use induction hypotheses there to show h(n+1) is divisible by (n+1)!, but in reality, I was just expanding the binomial theorem and reorganizing, the induction wasn't needed there. I could not find a way to combine both induction and the binomial property in the same proof, I am wondering if the linked proof is enough to solve my question, or do I have to do something to make use of the $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$ property. If this is not the right approach then how to prove the theorem making use of both induction and the binomial property?