There are $n$ balls and $m$ bins and every ball is placed independently and uniformly at random into a bin. I'm trying to show that there exists a constant $c$ such that, if $m=c\sqrt{n}$ then with probability $\geqslant 1/2$, no two balls will collide into the same bin, for sufficiently large $n$. (In textbooks I found the solution for upper bounding this probability.)
Let $A_i$ be the event that the $i$-th ball does not collide with balls $1,\dots,i-1$.
So essentially I want to lower bound the probability $ p = P\left[\ \bigcap_{i=2}^m A_i\ \right] = \prod_{i=2}^m\left( 1 - \frac{i-1}{n}\right). $ Assuming that $n \geqslant 2m$, we can use the bound $1-x\geqslant e^{-x-x^2}$ (for $x\leqslant \frac{1}{2}$). This gives $ p\geqslant \exp\left(\sum_{i=2}^m\left(-\frac{(i-1)^2}{n^2}-\frac{i-1}{n}\right)\right) = \exp\left( \frac{-(m-1) m (2 m+3 n-1)}{6 n^2} \right). $ But now I'm stuck, because setting $m=\Theta(\sqrt{n})$, doesn't seem to yield constant probability. What am I missing?