Chapter 4, Problem 4.62P

Consider n, independent trials, each of which results in one of the outcomes 1.. . ..k with respective probabilities $p_1,\mathrm{...},p_k,{\displaystyle {\sum }_{i=1}^k{p}_{i}1}$. Show that if all the $p_i$ are small, then the probability that no trial outcome occurs more than once is approximately equal to $\exp \left(-n\left(n-1\right){\displaystyle {\sum }_i\frac{p_i^2}{2}}\right)$.