Statement about divisibility

Question

Let's consider such function:

$$f(N) = 1^1\cdot 2^2\cdot 3^3 \dots (N-1)^{N-1}\cdot N^N.$$

Does the expression $$\frac{f(N)}{f(r)\cdot f(N-r)}$$

is always integer? Can you give me any hint about how to check it?

Answer 1

Answer: Yes, it is always integer.

Hint: write $f(n)=\frac{n!^{n+1}}{n!(n-1)!\cdots1!}$.

Full solution: (It's a matter of puzzling with the factors)

WLOG assume $r\leqslant n-r$. We have $$\begin{align*}\frac{f(n)}{f(r)f(n-r)}&=\frac{n!^{n+1}}{n!(n-1)!\cdots1!}\frac{r!\cdots1!}{r!^{r+1}}\frac{(n-r)!\cdots1!}{(n-r)!^{n-r+1}}\\&=\left(\frac{n!}{r!(n-r)!}\right)^{n-r+1}\cdot r!^{n-2r}\cdot\frac{n!^{r}}{n!\cdots(n-r+1)!}\cdot\frac{(n-r)!\cdots1!}{(n-r)!\cdots1!}\cdot r!\cdots1!\end{align*}$$