So after looking at the factorial formula and learning about product notation, I recognized this relation between them: $$\prod_{n=1}^kn=k!$$ And after fooling around and doing some trial and error, I came up with a similar relation for permutations: $$\prod_{n=r}^{k}n = (_kP_{k-r+1}) = P(k,k-r+1)$$
But when it came to combinations, the only formula I was able to create was this: $$\frac{1}{(k-r+1)!}\prod_{n=r}^{k}n = (_kC_{k-r+1}) = {k \choose k-r+1}$$ So my question is are there any formulas for combinations that only involve a product notation? That is, an equation using product notation without having to multiply out front by $\frac{1}{(k-r+1)!}$? Thanks for any answers