I have to prove that if $\displaystyle n_{1}+n_{2}=n $ and$\displaystyle \frac{n_{1}}{n}=p$, then prove $\displaystyle \binom rk (p-\frac{k}{n})^k (q-\frac{r-k}{n})^{r-k}<\frac{\binom{r}{k}\binom{n-r}{n_{1}-k}}{\binom {n}{n_{1}}}<\binom rk p^k q^{r-k}(1-\frac{r}{n})^{-r}$

I have tried and simplified it to show that this reduces to proving $\displaystyle (n_{1}-k)^k (n-n_{1}-r+k)^{r-k}<\frac{\binom{r}{k}\binom{n-r}{n_{1}-k}}{\binom {n}{n_{1}}}<{n_{1}}^k(n-n_{1})^{r-k}(n-r)^{-r}$

But I cant see how to proceed further to prove this.