Is the product of two factorial always smaller than the factorial of the sum?

Question

Let's imagine that we have two positive integers $k_1$ and $k_2$ so that $k = k_1 + k_2$. Can we assume that for any pair of $k_1$, $k_2$: $$ k_1! k_2! \leq k! \quad \text{?} $$ Intuitively, I would say yes, but I'm looking for a proof.

Answer 1

Yes,

$$k! = (k_1 + k_2)! = k_1! (k_1+1) \dots (k_1 + k_2) > k_1! \cdot 1 \dots k_2$$

Because $k_1 + i > i$ for natural $i$ and positive $k_1$.

Answer 2

Yes. Let $k' := k!/k_1!$. Then $k'$ is a product of $k_2$ integers (from $k_1 + 1$ to $k$), all larger than $k_1 \geq 0$. In particular, $k'$ is larger than $k_2!$, and so $k! = k_1 ! k' \geq k_1 ! k_2 !$.