The product of $r$ consecutive positive integers is divisible by

Step 1 (Formulate the Product): Let the product of $r$ consecutive positive integers be represented by:

where $n\ge r$ is the largest of these consecutive integers.

Step 2 (Relate to Binomial Coefficients): The definition of the binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ is:

Step 3 (Integrality Property): A fundamental mathematical property is that the binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ is always an integer for any positive integers $n\ge r$.
Since $\frac{P}{r!}=\text{Integer}=K$:

This proves that the product $P$ of $r$ consecutive positive integers is always divisible by $r!$.

Step 4 (Conclusion): The product is divisible by r!.